Felix' Math Place (Posts about topological argument.)
https://math.fontein.de/tag/topological-argument.atom
2019-11-17T10:38:16Z
Felix Fontein
Nikola
Diagonalizable Matrices.
https://math.fontein.de/2010/01/29/diagonalizable-matrices/
2010-01-29T04:47:39+01:00
2010-01-29T04:47:39+01:00
Felix Fontein
<div>
<p>
Today, during a lecture, we were posed the question whether <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/HNXqh7dpPSyWap4OW5elr.ZcZ2W_N5.0mJYiRQ.svgz" alt="D_n(K)" title="D_n(K)"></span>, the set of diagonalizable <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="12" src="https://math.fontein.de/formulae/n2t_qPF6pMPzbL.ASsuQSRtq3timjo4OYJy5cg.svgz" alt="n \times n" title="n \times n"></span> matrices over an algebraically closed field <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, is Zariski-open, i.e. open in the <a href="https://en.wikipedia.org/wiki/Zariski_topology">Zariski topology</a>. This would imply that in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/cHGmSnJY7aZUGhbREffNRBwCn79nLbwe1d7xdQ.svgz" alt="K = \C" title="K = \C"></span>, the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/rigO6MFb.mKOxNadI90EBvPHfeCRplEo3E69JQ.svgz" alt="D_n(M)" title="D_n(M)"></span> would be open and dense in <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="19" src="https://math.fontein.de/formulae/JA6HS9OyBqQnmXJIUi7CyCPUJ.yWgGuro6qSKw.svgz" alt="M_n(K) = \R^{n \times n}" title="M_n(K) = \R^{n \times n}"></span> in the <a href="https://en.wikipedia.org/wiki/Standard_topology#Topology_of_Euclidean_space">standard (Euclidean) topolgy</a>.
</p>
<p>
Unfortunately, the answer turns out to be “no” for the case <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/cHGmSnJY7aZUGhbREffNRBwCn79nLbwe1d7xdQ.svgz" alt="K = \C" title="K = \C"></span> (as well as <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/6LMDm8s8_La_eisR.EwfW4CG3W0YM2lSNqLc_Q.svgz" alt="K = \R" title="K = \R"></span>):
</p>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="14" src="https://math.fontein.de/formulae/y9JtZ2VqSEdReoO8vvhANI90r_lEdpV6KR4sTA.svgz" alt="n \ge 2" title="n \ge 2"></span>. Consider the matrix
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="227" height="76" src="https://math.fontein.de/formulae/P8aUGWmYnS3HwWf8unPCtjI0EbX_KbqeJ2ivcg.svgz" alt="A := \Matrix{ 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 } \in D_n(\C)," title="A := \Matrix{ 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 } \in D_n(\C),">
</div>
<p>
as well as the sequence
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="335" height="142" src="https://math.fontein.de/formulae/13CkvyOsH3YqUYh_n4oueU7lmrMg6YX9rYNqDw.svgz" alt="A_m := \Matrix{ 0 & 1/m & 0 & \cdots & 0 \\ \vdots & \ddots & 0 & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 0 \\
0 & \cdots & \cdots & \cdots & 0 } \in M_n(\C)." title="A_m := \Matrix{ 0 & 1/m & 0 & \cdots & 0 \\ \vdots & \ddots & 0 & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 0 \\
0 & \cdots & \cdots & \cdots & 0 } \in M_n(\C).">
</div>
<p>
Clearly, <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="15" src="https://math.fontein.de/formulae/0Mes4bQWoid0aTE3D94mmILasalQZhrL1Wr6eQ.svgz" alt="\lim_{m\to\infty} A_m = A" title="\lim_{m\to\infty} A_m = A"></span>. Assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/JKh57NxiP.4ZwsV6qxeBsHuwVxxL_mqlMHbLYw.svgz" alt="D_n(\C)" title="D_n(\C)"></span> is open in <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="18" src="https://math.fontein.de/formulae/uTMc6Fzpdg1qcYP6mloBktbsiraJgVQ3FCRrFQ.svgz" alt="M_n(\C)" title="M_n(\C)"></span>; then we must have <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/sk2sCiQAd7ggdQx6OYa4i7bL9bSmPp66CQOYjw.svgz" alt="A_m \in D_n(\C)" title="A_m \in D_n(\C)"></span> for almost all <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="13" src="https://math.fontein.de/formulae/lZq6w7FC4RcOgrJ2Nfd2.bCqYfEwb5lFwKGBOg.svgz" alt="m \in \N" title="m \in \N"></span>. But <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="15" src="https://math.fontein.de/formulae/fkONETxz.2Fx_odgmw8zf0CGW1thy.._KvINGg.svgz" alt="m A_m" title="m A_m"></span> is in <a href="https://en.wikipedia.org/wiki/Jordan_canonical_form">Jordan canonical form</a>, and clearly not diagonalizable; but this means that <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/ntHkGY0GVOV5bqQz2jjqB3n39_Gp.WQHayMuMg.svgz" alt="A_m \not\in D_n(\C)" title="A_m \not\in D_n(\C)"></span> for <strong>all</strong> <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="13" src="https://math.fontein.de/formulae/lZq6w7FC4RcOgrJ2Nfd2.bCqYfEwb5lFwKGBOg.svgz" alt="m \in \N" title="m \in \N"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/JKh57NxiP.4ZwsV6qxeBsHuwVxxL_mqlMHbLYw.svgz" alt="D_n(\C)" title="D_n(\C)"></span> is not open in <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="18" src="https://math.fontein.de/formulae/uTMc6Fzpdg1qcYP6mloBktbsiraJgVQ3FCRrFQ.svgz" alt="M_n(\C)" title="M_n(\C)"></span>.
</p>
</div>
</div>
<p>
But nonetheless, <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/HNXqh7dpPSyWap4OW5elr.ZcZ2W_N5.0mJYiRQ.svgz" alt="D_n(K)" title="D_n(K)"></span> contains a Zariski-open subset of <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/AjVwGzh4zzesjZG3FdVeX124v679_g.SodNqZw.svgz" alt="M_n(K)" title="M_n(K)"></span> in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is algebraically closed (which implies that <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/JKh57NxiP.4ZwsV6qxeBsHuwVxxL_mqlMHbLYw.svgz" alt="D_n(\C)" title="D_n(\C)"></span> lies dense in <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="18" src="https://math.fontein.de/formulae/uTMc6Fzpdg1qcYP6mloBktbsiraJgVQ3FCRrFQ.svgz" alt="M_n(\C)" title="M_n(\C)"></span>). For that recall that <span class="inline-formula"><img class="img-inline-formula img-formula" width="210" height="18" src="https://math.fontein.de/formulae/WJBae1NwQ3KzTOhdU39ytjYdUklkQtc6JbZM4w.svgz" alt="\chi_A = \det(x E_n - A) \in K[x]" title="\chi_A = \det(x E_n - A) \in K[x]"></span> is the characteristic polynomial of <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="18" src="https://math.fontein.de/formulae/5zVqD2Deu41UkezIPp_j5c5iQXP0QjO.U1gOJA.svgz" alt="A \in M_n(K)" title="A \in M_n(K)"></span>.
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
Consider the set
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="338" height="18" src="https://math.fontein.de/formulae/hAoigJ01rrrALnICBOzQWYNZA756RsUHGvtcMQ.svgz" alt="V_n(K) := \{ A \in M_n(K) \mid \chi_A \text{ is squarefree } \}." title="V_n(K) := \{ A \in M_n(K) \mid \chi_A \text{ is squarefree } \}.">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="18" src="https://math.fontein.de/formulae/3NhSyA6GqY2tqw6q3OiITM3yqXVGHWmhxblp7A.svgz" alt="V_n(K) \subseteq M_n(K)" title="V_n(K) \subseteq M_n(K)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="18" src="https://math.fontein.de/formulae/8qERSA9QN_oUJ7BpOqVvfUKNVOhZMo4DeUze2w.svgz" alt="V_n(K)" title="V_n(K)"></span> is Zariski-open in <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/AjVwGzh4zzesjZG3FdVeX124v679_g.SodNqZw.svgz" alt="M_n(K)" title="M_n(K)"></span>. In fact, <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="18" src="https://math.fontein.de/formulae/8qERSA9QN_oUJ7BpOqVvfUKNVOhZMo4DeUze2w.svgz" alt="V_n(K)" title="V_n(K)"></span> is the complement of a hypersurface in <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/AjVwGzh4zzesjZG3FdVeX124v679_g.SodNqZw.svgz" alt="M_n(K)" title="M_n(K)"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Note that in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span> is squarefree, <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span> splits into distinct linear factors since <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is algebraically closed. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> has <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> distinct eigenvalues in <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> and therefore one obtains <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> linearly independent eigenvectors of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span>; i.e., <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> is diagonalizable over <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="18" src="https://math.fontein.de/formulae/3NhSyA6GqY2tqw6q3OiITM3yqXVGHWmhxblp7A.svgz" alt="V_n(K) \subseteq M_n(K)" title="V_n(K) \subseteq M_n(K)"></span>.
</p>
<p>
Now we show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="18" src="https://math.fontein.de/formulae/qXOWtaWJ8Zx8tjbi0L3.2E.f1RmgVmSlgsooPA.svgz" alt="M_n(K) \setminus V_n(K)" title="M_n(K) \setminus V_n(K)"></span> is a hypersurface in <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/AjVwGzh4zzesjZG3FdVeX124v679_g.SodNqZw.svgz" alt="M_n(K)" title="M_n(K)"></span>, i.e. there exists a polynomial <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="18" src="https://math.fontein.de/formulae/RZ1wC6p4IMohny832CT9.twf6ID.Wa6Cw.lXiA.svgz" alt="f \in K[x_{ij} \mid 1 \le i, j \le n]" title="f \in K[x_{ij} \mid 1 \le i, j \le n]"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="268" height="18" src="https://math.fontein.de/formulae/HAwN80QMlK1JhKpLQLYNBGhTFZgE0IwjyOIy_Q.svgz" alt="V_n(K) = \{ A \in M_n(K) \mid f(A) \neq 0 \}" title="V_n(K) = \{ A \in M_n(K) \mid f(A) \neq 0 \}"></span>. For that, consider the maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="208" height="18" src="https://math.fontein.de/formulae/IslvfcTr6qs3UdIAmflzy8dk28USBJz.kDH4_A.svgz" alt="f_0, \dots, f_{n-1} : M_n(K) \to K" title="f_0, \dots, f_{n-1} : M_n(K) \to K"></span> defined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="22" src="https://math.fontein.de/formulae/kmj9W57La9kUFNGoePkcTF.USFNhwDj2GNluEA.svgz" alt="\chi_A = x^n + \sum_{i=0}^{n-1} f_i(A) x^i" title="\chi_A = x^n + \sum_{i=0}^{n-1} f_i(A) x^i"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="18" src="https://math.fontein.de/formulae/5zVqD2Deu41UkezIPp_j5c5iQXP0QjO.U1gOJA.svgz" alt="A \in M_n(K)" title="A \in M_n(K)"></span>. Obviously, these <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="16" src="https://math.fontein.de/formulae/tAXGS8UjI_jQ_x_GJXePp0Wdem_ejE1lHPqoZQ.svgz" alt="f_i" title="f_i"></span> must be polynomials. Next, consider the <a href="https://en.wikipedia.org/wiki/Discriminant#Discriminant_of_a_polynomial">discriminant</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="18" src="https://math.fontein.de/formulae/SUdfwqsZEvXkmeOymvJSBc3oll.QTOnfjTVtEg.svgz" alt="D(\chi_A)" title="D(\chi_A)"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span>; this is a polynomial expression in the coefficients of <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span>, i.e. in <span class="inline-formula"><img class="img-inline-formula img-formula" width="163" height="18" src="https://math.fontein.de/formulae/44mEzWKVlBQDs7gbF5cxXZGDir4302.wVOUYSQ.svgz" alt="1, f_0(A), \dots, f_{n-1}(A)" title="1, f_0(A), \dots, f_{n-1}(A)"></span>, whose value is zero if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span> is squarefree. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="197" height="18" src="https://math.fontein.de/formulae/SV2o.7BcKTiCQJOAoLUiMLyiYJB3D_l0ChnyWg.svgz" alt="A \in V_n(K) \Leftrightarrow D(\chi_A) \neq 0" title="A \in V_n(K) \Leftrightarrow D(\chi_A) \neq 0"></span>. Finally, <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/.083X56IAPeJSaw6XtJLBk9LHohTPzyJcctXdw.svgz" alt="f := D(\chi_A)" title="f := D(\chi_A)"></span> is a polynomial, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="268" height="18" src="https://math.fontein.de/formulae/HAwN80QMlK1JhKpLQLYNBGhTFZgE0IwjyOIy_Q.svgz" alt="V_n(K) = \{ A \in M_n(K) \mid f(A) \neq 0 \}" title="V_n(K) = \{ A \in M_n(K) \mid f(A) \neq 0 \}"></span> is Zariski-open in <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/AjVwGzh4zzesjZG3FdVeX124v679_g.SodNqZw.svgz" alt="M_n(K)" title="M_n(K)"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Note that the situation is different over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>:
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
In the standard topology,
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="439" height="74" src="https://math.fontein.de/formulae/qky0dEXPy2JlD.pEVI_wLGbHyDTYMwSHtVv.6w.svgz" alt="& \overline{D_n(\R)} = \overline{D_n(\R) \cap V_n(\R)} \\
{}={} & \{ A \in M_n(\R) \mid A \text{ has only real eigenvalues } \} \\
{}={} & \{ A \in M_n(\R) \mid A \text{ has a Jordan canonical form over } \R \}." title="& \overline{D_n(\R)} = \overline{D_n(\R) \cap V_n(\R)} \\
{}={} & \{ A \in M_n(\R) \mid A \text{ has only real eigenvalues } \} \\
{}={} & \{ A \in M_n(\R) \mid A \text{ has a Jordan canonical form over } \R \}.">
</div>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> has at least one eigenvalue <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/jfypCeE7ybMPnhD.TcSCALIZRLBZckuutt6fcw.svgz" alt="\lambda \in \C" title="\lambda \in \C"></span> with imaginary part <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="16" src="https://math.fontein.de/formulae/wWfRbHyPgYNEKZvtIJ3GVNAvhrQuDO_aIGL7sw.svgz" alt="\Im \lambda \neq 0" title="\Im \lambda \neq 0"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/rZGr2U3oYUfmM7XlNFU7t96nM3C0rEOnKLX7Jw.svgz" alt="(A_m)_m" title="(A_m)_m"></span> is a sequence of matrices with <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="15" src="https://math.fontein.de/formulae/0Mes4bQWoid0aTE3D94mmILasalQZhrL1Wr6eQ.svgz" alt="\lim_{m\to\infty} A_m = A" title="\lim_{m\to\infty} A_m = A"></span>, each <span class="inline-formula"><img class="img-inline-formula img-formula" width="27" height="15" src="https://math.fontein.de/formulae/_qzMq2oVQZHwVuLBLnELT6jk6uwrBYVEX7akug.svgz" alt="A_m" title="A_m"></span> must have an eigenvalue <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="15" src="https://math.fontein.de/formulae/aC_d4DCeSfawmVeDfCgp8AJeDxXSvjCbZ6nj2w.svgz" alt="\lambda_m \in \C" title="\lambda_m \in \C"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="15" src="https://math.fontein.de/formulae/zFia2Lmonpn45rGuZq3xDACw32aAoTSwcRZ2HA.svgz" alt="\lim_{m\to\infty} \lambda_m = \lambda" title="\lim_{m\to\infty} \lambda_m = \lambda"></span>. But then, for infinitely many <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="8" src="https://math.fontein.de/formulae/Ln3FmQu8Rxbv_tjKkeMJZrLhqAFAWD18KRPi9w.svgz" alt="m" title="m"></span>, we must have <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="16" src="https://math.fontein.de/formulae/Gdigzy4q66AN.SLz7CwTwx1t.xXDhUwgghFNZQ.svgz" alt="\lambda_m \not\in \R" title="\lambda_m \not\in \R"></span> (since <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="18" src="https://math.fontein.de/formulae/620DoZeL1NsMrbpHid5d1TaVfr316u2xJgBk9A.svgz" alt="\C \setminus \R" title="\C \setminus \R"></span> is open), whence we cannot have <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/cQMcD6JCn3ReOM9Tb7W5m4QzONfVkpVN1Hzi.A.svgz" alt="A_m \in D_n(\R)" title="A_m \in D_n(\R)"></span> for infinitely many <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="8" src="https://math.fontein.de/formulae/Ln3FmQu8Rxbv_tjKkeMJZrLhqAFAWD18KRPi9w.svgz" alt="m" title="m"></span>. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> is not in the closure of <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/87JM3H6PTxJg9m9yp.b0ItXivj.V47.IGoQKdA.svgz" alt="D_n(\R) \cap V_n(\R)" title="D_n(\R) \cap V_n(\R)"></span>.
</p>
<p>
Now assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> has only real eigenvalues. Then there exist some <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/tsIhCcng3Qa80b4BC3DAnuSmkw5FaRmxCHOUxg.svgz" alt="T \in GL_n(\R)" title="T \in GL_n(\R)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="14" src="https://math.fontein.de/formulae/HwZNYWlgR7EJ1fsVuqpoLFtg7Yt34KH9syMHWA.svgz" alt="T^{-1} A T" title="T^{-1} A T"></span> in Jordan canonical form. By pertubing the diagonal elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="14" src="https://math.fontein.de/formulae/HwZNYWlgR7EJ1fsVuqpoLFtg7Yt34KH9syMHWA.svgz" alt="T^{-1} A T" title="T^{-1} A T"></span> slightly, we can obtain a sequence of matrices <span class="inline-formula"><img class="img-inline-formula img-formula" width="166" height="18" src="https://math.fontein.de/formulae/OM6lOrB1zEo_N6NBY0kSo.LkBbRT79buLJrVtw.svgz" alt="B_m \in V_n(\R) \cap D_n(\R)" title="B_m \in V_n(\R) \cap D_n(\R)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="17" src="https://math.fontein.de/formulae/cTrb4TI1zeOBIUhXp9JY2FxeEYxtIjNK0NTt_w.svgz" alt="\lim_{m \to \infty} B_m \to T^{-1} A T" title="\lim_{m \to \infty} B_m \to T^{-1} A T"></span>. But then, <span class="inline-formula"><img class="img-inline-formula img-formula" width="178" height="17" src="https://math.fontein.de/formulae/Geb9d4RT4A1RzoR7KuOdsmw4fI9CzF8QuWyDhw.svgz" alt="\lim_{m\to\infty} T B_m T^{-1} = A" title="\lim_{m\to\infty} T B_m T^{-1} = A"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="210" height="19" src="https://math.fontein.de/formulae/sEXaud38ncG.7q8oy9DhjULaICmkxMFVLDVDyA.svgz" alt="T B_m T^{-1} \in V_n(\R) \cap D_n(\R)" title="T B_m T^{-1} \in V_n(\R) \cap D_n(\R)"></span> for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="13" src="https://math.fontein.de/formulae/lZq6w7FC4RcOgrJ2Nfd2.bCqYfEwb5lFwKGBOg.svgz" alt="m \in \N" title="m \in \N"></span>.
</p>
<p>
Note that this implies <span class="inline-formula"><img class="img-inline-formula img-formula" width="152" height="21" src="https://math.fontein.de/formulae/cu9GGNN_qOj8XHTivwGYpPMO1nqnR5dbY24EhA.svgz" alt="A \in \overline{V_n(\R) \cap D_n(\R)}" title="A \in \overline{V_n(\R) \cap D_n(\R)}"></span>; moreover, this also implies <span class="inline-formula"><img class="img-inline-formula img-formula" width="192" height="21" src="https://math.fontein.de/formulae/69t4bEtxdLfaIfI2DXhjNoeurLsbJPzWB7jq8Q.svgz" alt="D_n(\R) \subseteq \overline{D_n(\R) \cap V_n(\R)}" title="D_n(\R) \subseteq \overline{D_n(\R) \cap V_n(\R)}"></span>. Hence, the first two equalities hold. The third equality is standard.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Also note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="18" src="https://math.fontein.de/formulae/9uS0iowfEJK6PlfIfp0TgP1vntmZzAFdjtY4Dg.svgz" alt="V_n(\R) \not\subseteq D_n(\R)" title="V_n(\R) \not\subseteq D_n(\R)"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="12" src="https://math.fontein.de/formulae/reArLJOpe3J6eHLpP4vd0OMCtQxU46viESPUfA.svgz" alt="n > 1" title="n > 1"></span>, as the example
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="75" height="43" src="https://math.fontein.de/formulae/0Dk0Yb.1MbXpv8ixrv8zk4chbfiS98mIioZekw.svgz" alt="\Matrix{ 0 & 1 \\ -1 & 0 }" title="\Matrix{ 0 & 1 \\ -1 & 0 }">
</div>
<p>
(which is diagonalizable over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>, with eigenvalues <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="13" src="https://math.fontein.de/formulae/zrKQfBcriayKtmo0QIpWfy_RxwpuX.u7NsSZHA.svgz" alt="\pm i" title="\pm i"></span>) shows. So what about <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="21" src="https://math.fontein.de/formulae/_HC1OoIf7ZhNkoSc2Lufi5Gr41dyrkIgk5xfWw.svgz" alt="\overline{V_n(\R)}" title="\overline{V_n(\R)}"></span>? In fact, as in the case of <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/cHGmSnJY7aZUGhbREffNRBwCn79nLbwe1d7xdQ.svgz" alt="K = \C" title="K = \C"></span>, it turns out that <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="21" src="https://math.fontein.de/formulae/_HC1OoIf7ZhNkoSc2Lufi5Gr41dyrkIgk5xfWw.svgz" alt="\overline{V_n(\R)}" title="\overline{V_n(\R)}"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="18" src="https://math.fontein.de/formulae/MEN3Zs5QbHPv.24f_.CvhIEiCT2lVVE1nmKMBw.svgz" alt="M_n(\R)" title="M_n(\R)"></span>.
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
We have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="128" height="21" src="https://math.fontein.de/formulae/JMFe5JUDkgoNQIrhzJ7M0OTMbVyIBa53YAt5iQ.svgz" alt="\overline{V_n(\R)} = M_n(\R)." title="\overline{V_n(\R)} = M_n(\R).">
</div>
</div>
</div>
<p>
For the proof, we need a little lemma.
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="256" height="18" src="https://math.fontein.de/formulae/Zo9grTpq6yhlmdw06U8KfaVJ1HAkH7uWmu6pRA.svgz" alt="S := \{ f \in \R[x] \mid f \text{ is squarefree } \}" title="S := \{ f \in \R[x] \mid f \text{ is squarefree } \}"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="20" src="https://math.fontein.de/formulae/s_ZjTlO6O0g0gspbHsa1WkxuvCuwDEOvUIuttw.svgz" alt="\overline{S} = \R[x]" title="\overline{S} = \R[x]"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="65" height="18" src="https://math.fontein.de/formulae/L2mUX7NxcDn4ArD8WZc1D8Ibq9T0aANMYs3KWA.svgz" alt="f \in \R[x]" title="f \in \R[x]"></span> be an arbitrary polynomial. Write <span class="inline-formula"><img class="img-inline-formula img-formula" width="106" height="20" src="https://math.fontein.de/formulae/O7CJsoyKZeU23Ohma.sWhvoWHON3VBBu5EibEQ.svgz" alt="f = \lambda \prod_{i=1}^n p_i" title="f = \lambda \prod_{i=1}^n p_i"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="13" src="https://math.fontein.de/formulae/OjYHyCuXSEdYG996cZL60mQD1U_cYVHc4rZCVQ.svgz" alt="\lambda \in \R^*" title="\lambda \in \R^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="18" src="https://math.fontein.de/formulae/cCodUFvjEcQ89jk4py7.aKu5pJ0MyZ8.pdl08w.svgz" alt="p_i \in \R[x]" title="p_i \in \R[x]"></span> is irreducible and monic, <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="14" src="https://math.fontein.de/formulae/FpZ4n1WCX1xSEeoMKKG9Jn.6P1n9K4u5Fncztw.svgz" alt="1 \le i \le n" title="1 \le i \le n"></span>. Now the coefficients of all <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="11" src="https://math.fontein.de/formulae/GtawJPCqwEmC0jA1vxn8euVlhIm4ArvuyTkdYg.svgz" alt="p_i" title="p_i"></span>'s (except the highest coefficients) are a finite set in <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="20" src="https://math.fontein.de/formulae/ECyyS4W0C1t27hcJ_v0alB6vnumnN1rj2G36Yw.svgz" alt="d := \sum_{i=1}^n \deg p_i" title="d := \sum_{i=1}^n \deg p_i"></span> elements, whence there exists sequences <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="24" src="https://math.fontein.de/formulae/iXQlzPuL38fpqnseGyc6yEjLpbFeMdEi35GclA.svgz" alt="(a_1^{(m)}, \dots, a_d^{(m)})" title="(a_1^{(m)}, \dots, a_d^{(m)})"></span> with pairwise distinct <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="24" src="https://math.fontein.de/formulae/Yl0IymrpUU7r_Nwm2adQDRl1OpyZJJSikoRyOg.svgz" alt="a_1^{(m)}, \dots, a_d^{(m)}" title="a_1^{(m)}, \dots, a_d^{(m)}"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="23" src="https://math.fontein.de/formulae/TwNbwjij9DgTGKaOIo8_W6qHJPvIEWabWKVmiw.svgz" alt="\lim a_i^{(m)}" title="\lim a_i^{(m)}"></span> converges to one coefficent of one <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="13" src="https://math.fontein.de/formulae/uBQZAWAfXQuZSPpqQeFTkOC8a8gKfNGA_BziOw.svgz" alt="p_j" title="p_j"></span>. In particular, we can construct monic polynomials <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="23" src="https://math.fontein.de/formulae/0j7oCO_brAS_3gw2uTN2.9_zScEKRt3an8pG4g.svgz" alt="p_i^{(m)} \in \R[x]" title="p_i^{(m)} \in \R[x]"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="23" src="https://math.fontein.de/formulae/4p8k2oshdot23.fMxUyvYx5RCF_N7eL1AJ2wVA.svgz" alt="\deg p_i^{(m)} = \deg p_i" title="\deg p_i^{(m)} = \deg p_i"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="141" height="23" src="https://math.fontein.de/formulae/HU3VijNaoFAerjIPWyn1.VMZvDlBz4snIXmqOQ.svgz" alt="\lim_{m\to\infty} p_i^{(m)} = p_i" title="\lim_{m\to\infty} p_i^{(m)} = p_i"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="90" height="26" src="https://math.fontein.de/formulae/fu4IzQB_UZw5feB8b8SxumO3QJgB60fm0tsiag.svgz" alt="p_i^{(m)} \neq p_j^{(m)}" title="p_i^{(m)} \neq p_j^{(m)}"></span> for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="38" height="16" src="https://math.fontein.de/formulae/sdNdxAE_xOEv38foNeA4HVOc.un.3BWONUcHgw.svgz" alt="i \neq j" title="i \neq j"></span>. Even more, we can make sure that every <span class="inline-formula"><img class="img-inline-formula img-formula" width="33" height="23" src="https://math.fontein.de/formulae/xUpFyJWG_Ob1TeyHCrdVUQK5AONW5Fl_2sN8LQ.svgz" alt="p_i^{(m)}" title="p_i^{(m)}"></span> is irreducible; this enforces that <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="24" src="https://math.fontein.de/formulae/0VY9u9thisd9gS6ejE0HH9yUZVBl5S1vLQxk4A.svgz" alt="f_m := \prod_{i=1}^n p_i^{(m)}" title="f_m := \prod_{i=1}^n p_i^{(m)}"></span> is squarefree, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="16" src="https://math.fontein.de/formulae/UfcXlGrQ411ORG1CcvoEpeMRIedrV2bc4TkLCw.svgz" alt="f_m \in S" title="f_m \in S"></span>. Therefore, we found a sequence in <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> converging to <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="19" src="https://math.fontein.de/formulae/GnnVf_.ffL065lxB7DRiimbflqdrfgZKHafBjw.svgz" alt="f \in \overline{S}" title="f \in \overline{S}"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof (Proof of the Proposition).
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="18" src="https://math.fontein.de/formulae/hHnWyReHfOrk35aFi2DyQ0VvqlfAeBDLcMucGA.svgz" alt="A \in M_n(\R)" title="A \in M_n(\R)"></span> whose characteristic polynomial <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="11" src="https://math.fontein.de/formulae/qSkXMn9VZpB11t74tSuxTfAY52wZHdGVY38U8Q.svgz" alt="\chi_A" title="\chi_A"></span> can be written as <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="22" src="https://math.fontein.de/formulae/RzuqY.4MgSS8qu56oQ9eXMDSTSseRzpyoaHtBw.svgz" alt="\prod_{i=1}^t p_i" title="\prod_{i=1}^t p_i"></span>, with not necessarily distinct, but monic and irreducible polynomials <span class="inline-formula"><img class="img-inline-formula img-formula" width="129" height="18" src="https://math.fontein.de/formulae/sq2.2zusN2AvMFkTIucupv5Zn5.0aoKRfz8H_w.svgz" alt="p_1, \dots, p_n \in \R[x]" title="p_1, \dots, p_n \in \R[x]"></span>. There exists a matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/tsIhCcng3Qa80b4BC3DAnuSmkw5FaRmxCHOUxg.svgz" alt="T \in GL_n(\R)" title="T \in GL_n(\R)"></span> with
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="228" height="76" src="https://math.fontein.de/formulae/dMxst6m5Lj5WPUcieifZGfcoPcf2352AD0xWLw.svgz" alt="T^{-1} A T = \Matrix{ C_{p_1} & & 0 \\ & \ddots & \\ 0 & & C_{p_t} }," title="T^{-1} A T = \Matrix{ C_{p_1} & & 0 \\ & \ddots & \\ 0 & & C_{p_t} },">
</div>
<p>
where <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="17" src="https://math.fontein.de/formulae/Uu7efL.cD1nJq0QevMZdG26s_yGps3stfG7jwg.svgz" alt="C_{p_i}" title="C_{p_i}"></span> is the <a href="https://en.wikipedia.org/wiki/Companion_matrix">companion matrix</a> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="11" src="https://math.fontein.de/formulae/GtawJPCqwEmC0jA1vxn8euVlhIm4ArvuyTkdYg.svgz" alt="p_i" title="p_i"></span>; this is a <a href="https://en.wikipedia.org/wiki/Frobenius_normal_form">Frobenius normal form</a> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span>. Now we can find a sequence of squarefree polynomials <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="23" src="https://math.fontein.de/formulae/0j7oCO_brAS_3gw2uTN2.9_zScEKRt3an8pG4g.svgz" alt="p_i^{(m)} \in \R[x]" title="p_i^{(m)} \in \R[x]"></span> such that for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="8" src="https://math.fontein.de/formulae/Ln3FmQu8Rxbv_tjKkeMJZrLhqAFAWD18KRPi9w.svgz" alt="m" title="m"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="106" height="23" src="https://math.fontein.de/formulae/2U1gqbz.XsWLuVISlmakI2VNtkVLAwTaObEGhg.svgz" alt="p_1^{(m)}, \dots, p_t^{(m)}" title="p_1^{(m)}, \dots, p_t^{(m)}"></span> are pairwise coprime, and that <span class="inline-formula"><img class="img-inline-formula img-formula" width="141" height="23" src="https://math.fontein.de/formulae/HU3VijNaoFAerjIPWyn1.VMZvDlBz4snIXmqOQ.svgz" alt="\lim_{m\to\infty} p_i^{(m)} = p_i" title="\lim_{m\to\infty} p_i^{(m)} = p_i"></span>. Then set
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="351" height="85" src="https://math.fontein.de/formulae/9EaSGPNvSFrbXCgYfb73nzWMKQ..gjxM18NfAg.svgz" alt="A_m := T \Matrix{ C_{p_1^{(m)}} & & 0 \\ & \ddots & \\ 0 & & C_{p_t^{(m)}} } T^{-1} \in M_n(\R);" title="A_m := T \Matrix{ C_{p_1^{(m)}} & & 0 \\ & \ddots & \\ 0 & & C_{p_t^{(m)}} } T^{-1} \in M_n(\R);">
</div>
<p>
clearly, <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="15" src="https://math.fontein.de/formulae/0Mes4bQWoid0aTE3D94mmILasalQZhrL1Wr6eQ.svgz" alt="\lim_{m\to\infty} A_m = A" title="\lim_{m\to\infty} A_m = A"></span>. Moreover, the characteristic polynomial of <span class="inline-formula"><img class="img-inline-formula img-formula" width="27" height="15" src="https://math.fontein.de/formulae/_qzMq2oVQZHwVuLBLnELT6jk6uwrBYVEX7akug.svgz" alt="A_m" title="A_m"></span> is given by <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="24" src="https://math.fontein.de/formulae/Mpeq5QjXrU2hYebg4_2qlJbizivnv1uvkhEHhg.svgz" alt="\prod_{i=1}^t p_i^{(m)}" title="\prod_{i=1}^t p_i^{(m)}"></span>, i.e. it is squarefree by choice of the <span class="inline-formula"><img class="img-inline-formula img-formula" width="33" height="23" src="https://math.fontein.de/formulae/xUpFyJWG_Ob1TeyHCrdVUQK5AONW5Fl_2sN8LQ.svgz" alt="p_i^{(m)}" title="p_i^{(m)}"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="95" height="18" src="https://math.fontein.de/formulae/p5CdT5sbayClmsEmiV4Vs4.bj86vlr4Sy0mR0Q.svgz" alt="A_m \in V_n(\R)" title="A_m \in V_n(\R)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="21" src="https://math.fontein.de/formulae/mfdcy3rCMlcCVMCl4QsxnfNy4CcIrT7qkVbUvA.svgz" alt="A \in \overline{V_n(\R)}" title="A \in \overline{V_n(\R)}"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
</div>
A Topological Proof of the Cayley-Hamilton Theorem over all Commutative Unitary Rings.
https://math.fontein.de/2009/05/04/a-topological-proof-of-the-cayley-hamilton-theorem-over-all-commutative-unitary-rings/
2009-05-04T06:52:19+02:00
2009-05-04T06:52:19+02:00
Felix Fontein
<div>
<p>
In this post, I want to present a very elegant proof of the Cayley-Hamilton Theorem which works over all commutative unitary <a href="https://en.wikipedia.org/wiki/Ring_%28mathematics%29">rings</a> by reducing to the case over the complex numbers, where a topological argument is used to reduce to the case of diagonalizable matrices. First of all, let us state the definitions and the theorem itself.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> be a commutative unitary ring and <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="15" src="https://math.fontein.de/formulae/_DA3Fjfkbj_6spDdCK3MJwKg203.nhdYfPUfhQ.svgz" alt="A \in R^{n \times n}" title="A \in R^{n \times n}"></span> a <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="12" src="https://math.fontein.de/formulae/n2t_qPF6pMPzbL.ASsuQSRtq3timjo4OYJy5cg.svgz" alt="n \times n" title="n \times n"></span>-matrix over <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>. The <em>characteristic polynomial</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> is the polynomial <span class="inline-formula"><img class="img-inline-formula img-formula" width="212" height="18" src="https://math.fontein.de/formulae/QDHTAexvzJ1Qg35uv2kyIE2hd3QYRa_FHECjrA.svgz" alt="\chi_A := \det(x E_n - A) \in
R[x]" title="\chi_A := \det(x E_n - A) \in
R[x]"></span>.
</p>
</div>
</div>
<p>
Then the theorem says:
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
<a name="cayleyhamiltonthm"></a>
Theorem (Cayley-Hamilton).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> be a commutative unitary ring and <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="15" src="https://math.fontein.de/formulae/_DA3Fjfkbj_6spDdCK3MJwKg203.nhdYfPUfhQ.svgz" alt="A \in R^{n \times n}" title="A \in R^{n \times n}"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/hIyHTKO9mQ4kswRSfoSMpc1QZKR9c2jLqJFx6Q.svgz" alt="\chi_A(A) = 0" title="\chi_A(A) = 0"></span>.
</p>
</div>
</div>
<p>
We first begin with a fascinating reduction argument, which I first saw in a lecture of <a href="http://www.math.ucla.edu/%20balmer/">Paul Balmer</a> at the <a href="http://www.ethz.ch/">ethz</a>:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
<a name="cayleyhamiltonreduction"></a>
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
The Theorem of Cayley-Hamilton holds over any commutative unitary ring if, and only if, it holds over the complex numbers.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Clearly, if the theorem holds for all rings, so it does for the special case <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="12" src="https://math.fontein.de/formulae/ug7s.Kf16VpV7xNnr2CZCPN2clovd34ckhb0xQ.svgz" alt="R = \C" title="R = \C"></span>. So assume that it holds for <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> be any commutative unitary ring and <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="15" src="https://math.fontein.de/formulae/_DA3Fjfkbj_6spDdCK3MJwKg203.nhdYfPUfhQ.svgz" alt="A \in R^{n \times n}" title="A \in R^{n \times n}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/cu_q5y8eXAHDWo8vQk4cqpr3EvnnfWz0vMr5KA.svgz" alt="A = (a_{ij})_{ij}" title="A = (a_{ij})_{ij}"></span>. Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="188" height="18" src="https://math.fontein.de/formulae/UW1v5kd83uPxyzA.JbeK4h1XOleuwHrGCPuuVQ.svgz" alt="S := \Z[x_{ij} \mid 1 \le i, j \le n]" title="S := \Z[x_{ij} \mid 1 \le i, j \le n]"></span> and consider the ring homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="16" src="https://math.fontein.de/formulae/U73PxTWNbDctFALv8Nzta35D0XvGgJYYpE5tUg.svgz" alt="\varphi : S \to R" title="\varphi : S \to R"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="186" height="18" src="https://math.fontein.de/formulae/oNkmTkDtGbwVSJ8XHLvIPzQWCAkxMzJVOBv54A.svgz" alt="f \mapsto f(a_{11}, a_{12}, \dots, a_{nn})" title="f \mapsto f(a_{11}, a_{12}, \dots, a_{nn})"></span>. Over <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span>, consider the matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="18" src="https://math.fontein.de/formulae/7EFzh.SN.Xjyf1C0WJ7GlQfQezscLm_WlIZuVQ.svgz" alt="B := (x_{ij})_{ij}" title="B := (x_{ij})_{ij}"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span> induces <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span>-algebra homomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/Gp4qJkW_DwpYpwhSXp9vzldZN.ws6Kv4KK2ArQ.svgz" alt="\varphi^* : S^{n \times n} \to R^{n \times n}" title="\varphi^* : S^{n \times n} \to R^{n \times n}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="18" src="https://math.fontein.de/formulae/Gr.jVqPp9S52.hI6jwPTIaeIGvG6YBEwfrajCg.svgz" alt="\varphi' : S[x] \to R[x]" title="\varphi' : S[x] \to R[x]"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/RbYfPo8khCEZt_Z3ctrqsQZcSejiUp.vhf86MQ.svgz" alt="\varphi^*(B) = A" title="\varphi^*(B) = A"></span>. Clearly, they satisfy <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/kKmcqpjvaCCYDJ1Il5AAC5WC6JbJa.UidAVfMg.svgz" alt="\varphi'(\chi_B) = \chi_A" title="\varphi'(\chi_B) = \chi_A"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/SJosdz71ZJ.wkTEfmbL4R7nwIKBmP3i2iQG4fQ.svgz" alt="\varphi^*(\chi_B(B)) = \chi_A(A)" title="\varphi^*(\chi_B(B)) = \chi_A(A)"></span>. Therefore, it suffices to prove <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/_o1Ac.L__tWwXeLHQ4oHMPKuqAu163yUa_G17A.svgz" alt="\chi_B(B) = 0" title="\chi_B(B) = 0"></span>.
</p>
<p>
Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> has infinite transcendence degree over <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="15" src="https://math.fontein.de/formulae/yic83_sv9OvU5TDbFAcLOkDAihqEngiRZV4YIQ.svgz" alt="\Q" title="\Q"></span> (otherwise, it could be countable), whence there exists an embedding <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="16" src="https://math.fontein.de/formulae/gRQ2ZxfC.770n_uU9jkMAjhiJM25BMuIh.dUPg.svgz" alt="\psi : S \to \C" title="\psi : S \to \C"></span>; simply choose <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="14" src="https://math.fontein.de/formulae/_toulBjHrvdLqHW.pCOwWk5LGqtdA9TaPOGgWA.svgz" alt="n^2" title="n^2"></span> algebraically independent elements in <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> and map the <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="13" src="https://math.fontein.de/formulae/THgCPLIeKcD3D2TAi2gVnIvgSyvOhQFRAMNLMw.svgz" alt="x_{ij}" title="x_{ij}"></span> to them. Again, we get maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/XufR5k3KikGRwxhjLlEjvzbLmVNsoHtEjbeiZg.svgz" alt="\psi^* : S^{n \times n} \to \C^{n \times n}" title="\psi^* : S^{n \times n} \to \C^{n \times n}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="18" src="https://math.fontein.de/formulae/5tGyGEkCZ4cUYWqmRhW9mKTUIayDUjpCM2lvBg.svgz" alt="\psi' : S[x] \to \C[x]" title="\psi' : S[x] \to \C[x]"></span> which are injective and satisfy <span class="inline-formula"><img class="img-inline-formula img-formula" width="129" height="20" src="https://math.fontein.de/formulae/cYPvnAceqk1cZO5_UZKqQSp9M0tkO7VcEBxcnA.svgz" alt="\psi'(\chi_B) = \chi_{\psi^*(B)}" title="\psi'(\chi_B) = \chi_{\psi^*(B)}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="363" height="20" src="https://math.fontein.de/formulae/vM8O1PnG5HNRPYMewKKSWcS5JmGvFn.uTw7L6Q.svgz" alt="\chi_{\psi^*(B)}(\psi^*(B)) = \psi'(\chi_B)(\psi^*(B)) = \psi^*(\chi_B(B))" title="\chi_{\psi^*(B)}(\psi^*(B)) = \psi'(\chi_B)(\psi^*(B)) = \psi^*(\chi_B(B))"></span>. But by assumption, Cayley-Hamilton holds over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="20" src="https://math.fontein.de/formulae/w2DCGfQ.nLX4i5V8Emo__JPslR920fsmnMbSRw.svgz" alt="\chi_{\psi^*(B)}(\psi^*(B)) = 0" title="\chi_{\psi^*(B)}(\psi^*(B)) = 0"></span>. Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="16" src="https://math.fontein.de/formulae/iWLO2uPupJFC98ARlpzIh4I_HTtAwj8hygcRAQ.svgz" alt="\psi^*" title="\psi^*"></span> is injective, <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/_o1Ac.L__tWwXeLHQ4oHMPKuqAu163yUa_G17A.svgz" alt="\chi_B(B) = 0" title="\chi_B(B) = 0"></span>, which implies <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/hIyHTKO9mQ4kswRSfoSMpc1QZKR9c2jLqJFx6Q.svgz" alt="\chi_A(A) = 0" title="\chi_A(A) = 0"></span> as mentioned above.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now we can concentrate on showing the Theorem of Cayley-Hamilton for the complex numbers. We begin with a special case, namely the diagonalizable matrices.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
A matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="15" src="https://math.fontein.de/formulae/_DA3Fjfkbj_6spDdCK3MJwKg203.nhdYfPUfhQ.svgz" alt="A \in R^{n \times n}" title="A \in R^{n \times n}"></span> is said to be <em>diagonalizable</em> if there exists an invertible matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/mgLTveGiO4ievOLgvTY5YDE_FNN9CGHD409oDQ.svgz" alt="T \in GL_n(R)" title="T \in GL_n(R)"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="405" height="109" src="https://math.fontein.de/formulae/z9svZEUdWWfaRSA.kwcxVwDvr0ajmc9SGUGfFQ.svgz" alt="T^{-1} A T = \Matrix{ \lambda_1 & 0 & \cdots & 0 \\ 0 &
\lambda_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \lambda_n } =:
diag(\lambda_1, \dots, \lambda_n)" title="T^{-1} A T = \Matrix{ \lambda_1 & 0 & \cdots & 0 \\ 0 &
\lambda_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \lambda_n } =:
diag(\lambda_1, \dots, \lambda_n)">
</div>
<p>
for <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="16" src="https://math.fontein.de/formulae/EuzUKRIr.smLVdkcQa90uxgeAHN3vFwtsqrCyg.svgz" alt="\lambda_1, \dots, \lambda_n \in R" title="\lambda_1, \dots, \lambda_n \in R"></span>.
</p>
</div>
</div>
<p>
We then have:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
The Theorem of Cayley-Hamilton holds for diagonalizable matrices.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
We first assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="162" height="18" src="https://math.fontein.de/formulae/_EzhQC7GBK1oZ9LZPr6Y_GtN_2OLh2KfF1btrA.svgz" alt="A = diag(\lambda_1, \dots, \lambda_n)" title="A = diag(\lambda_1, \dots, \lambda_n)"></span>. Then one gets <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="20" src="https://math.fontein.de/formulae/aL1wVuAEIJzRc99ursC6R.Z2yANeuYLQCf_Vgg.svgz" alt="\chi_A = \prod_{i=1}^n
(x - \lambda_i)" title="\chi_A = \prod_{i=1}^n
(x - \lambda_i)"></span>, and since
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="520" height="18" src="https://math.fontein.de/formulae/LxcDUAGoOJ3p9dDLCd65bKIqr.H5M4qOxzW0Qw.svgz" alt="(A - \lambda_i E_n) = diag(\lambda_1 - \lambda_i, \dots, \lambda_{i-1}
- \lambda_i, 0, \lambda_{i+1} - \lambda_i, \dots, \lambda_n - \lambda_i)" title="(A - \lambda_i E_n) = diag(\lambda_1 - \lambda_i, \dots, \lambda_{i-1}
- \lambda_i, 0, \lambda_{i+1} - \lambda_i, \dots, \lambda_n - \lambda_i)">
</div>
<p>
one gets <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/hIyHTKO9mQ4kswRSfoSMpc1QZKR9c2jLqJFx6Q.svgz" alt="\chi_A(A) = 0" title="\chi_A(A) = 0"></span>.
</p>
<p>
Now assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> is diagonalizable, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/mgLTveGiO4ievOLgvTY5YDE_FNN9CGHD409oDQ.svgz" alt="T \in GL_n(R)" title="T \in GL_n(R)"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="207" height="19" src="https://math.fontein.de/formulae/2KVt3D39ie_511t6JH3CUGmAViDN9q6l28xMNQ.svgz" alt="T^{-1} A T = diag(\lambda_1,
\dots, \lambda_n)" title="T^{-1} A T = diag(\lambda_1,
\dots, \lambda_n)"></span>. Clearly, <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="19" src="https://math.fontein.de/formulae/yYI9ADHhdVH12qMzJxkfqXXV8_xgD35sydDe0A.svgz" alt="\det T^{-1} = (\det T)^{-1}" title="\det T^{-1} = (\det T)^{-1}"></span> and, therefore,
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="475" height="48" src="https://math.fontein.de/formulae/mIj5ybPm3Zscr4ZCqU6yw3.zVa6vA4gRnlW_HQ.svgz" alt="\chi_A ={} & \det(x E_n - A) = \det T^{-1} \cdot \det(x E_n - A) \cdot \det T \\
{}={} & \det (T^{-1} (x E_n - A) T) = \det(x E_n - T^{-1} A T) = \chi_{T^{-1} A T}." title="\chi_A ={} & \det(x E_n - A) = \det T^{-1} \cdot \det(x E_n - A) \cdot \det T \\
{}={} & \det (T^{-1} (x E_n - A) T) = \det(x E_n - T^{-1} A T) = \chi_{T^{-1} A T}.">
</div>
<p>
Now write <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="20" src="https://math.fontein.de/formulae/WWKEEksbsvzW7AwPrQe4o21DLuoxO3RxfNPo7Q.svgz" alt="\chi_A = \sum_{i=0}^n a_i x^i" title="\chi_A = \sum_{i=0}^n a_i x^i"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="15" src="https://math.fontein.de/formulae/YJE9ORNMVIMGpWSMLn67sTaKh7L84jQD5wX29A.svgz" alt="a_i \in R" title="a_i \in R"></span>. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="493" height="52" src="https://math.fontein.de/formulae/yB3ZzFf5FgNJ69pNlfCEkCy4XMeCjGRt9fVnDQ.svgz" alt="T^{-1} \chi_A(A) T = \sum_{i=0}^n
a_i T^{-1} A^i T = \sum_{i=0}^n a_i (T^{-1} A T)^i = \chi_A(T^{-1} A T)," title="T^{-1} \chi_A(A) T = \sum_{i=0}^n
a_i T^{-1} A^i T = \sum_{i=0}^n a_i (T^{-1} A T)^i = \chi_A(T^{-1} A T),">
</div>
<p>
whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="249" height="19" src="https://math.fontein.de/formulae/PQRGs0HvvuQOGAfXbR1.3jHL1B9WYB07Va.N.Q.svgz" alt="T^{-1} \chi_A(A) T
= \chi_{T^{-1} A T}(T^{-1} A T)" title="T^{-1} \chi_A(A) T
= \chi_{T^{-1} A T}(T^{-1} A T)"></span>. But now <span class="inline-formula"><img class="img-inline-formula img-formula" width="207" height="19" src="https://math.fontein.de/formulae/mQ19YL6BNNVrlLFzUBSfSwKd_lRGeS2n0jTh_w.svgz" alt="T^{-1} A T = diag(\lambda_1, \dots, \lambda_n)" title="T^{-1} A T = diag(\lambda_1, \dots, \lambda_n)"></span>, whence we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="19" src="https://math.fontein.de/formulae/ZRkNN.0J_m0SgsMfjfIYtLEHZDAXGY_YODIv_A.svgz" alt="T^{-1} \chi_A(A) T = 0" title="T^{-1} \chi_A(A) T = 0"></span> and, hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/hIyHTKO9mQ4kswRSfoSMpc1QZKR9c2jLqJFx6Q.svgz" alt="\chi_A(A) = 0" title="\chi_A(A) = 0"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
We now get to the main piece of proving Cayley-Hamilton over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
<a name="diagmatricesdenselemma"></a>
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Endow <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="14" src="https://math.fontein.de/formulae/AGrdSL62ngoSFRvJP1JsZrPVccXNEtq4MbZs5Q.svgz" alt="\C^{n \times n}" title="\C^{n \times n}"></span> with the Euclidean topology and consider the set
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="293" height="19" src="https://math.fontein.de/formulae/SOOWywgNmUbwlMHuRsWK2EtYkP2mS0HpLpV9zg.svgz" alt="D := \{ A \in \C^{n \times n}
\mid A \text{ diagonalizable } \}." title="D := \{ A \in \C^{n \times n}
\mid A \text{ diagonalizable } \}.">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/.1gXmU_PRktx2Rs2BFoRNw.Q_4Ub6YJ34UcmGw.svgz" alt="D" title="D"></span> is dense in <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="14" src="https://math.fontein.de/formulae/AGrdSL62ngoSFRvJP1JsZrPVccXNEtq4MbZs5Q.svgz" alt="\C^{n \times n}" title="\C^{n \times n}"></span>.
</p>
</div>
</div>
<p>
For this proof, we need two facts from linear algebra:
</p>
<ul class="item-level-1">
<li>Every matrix over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> is equivalent to a <a href="https://de.wikipedia.org/wiki/Trigonalisierung">triagonal matrix</a>; this can be done if, and only if, the characteristic polynomial of the matrix splits into linear factors. But, by the <a href="https://math.fontein.de/2009/05/04/fundamental-theorem-of-algebra/">Fundamental Theorem of Algebra</a>, this is always the case over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>.</li>
<li>An <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="12" src="https://math.fontein.de/formulae/n2t_qPF6pMPzbL.ASsuQSRtq3timjo4OYJy5cg.svgz" alt="n \times n" title="n \times n"></span>-matrix with <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> distinct eigenvalues is diagonalizable.</li>
</ul>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="15" src="https://math.fontein.de/formulae/EeC2.bIXJd2Tfm7OiK.1Z8xFWDUjh09NZf5.ng.svgz" alt="A \in \C^{n \times n}" title="A \in \C^{n \times n}"></span> be an arbitrary matrix. Then there exists a matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/T19Lfj5dmkjvEoI.K8Xky6_EvxGFN3G.7DjbDg.svgz" alt="T \in GL_n(\C)" title="T \in GL_n(\C)"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="251" height="109" src="https://math.fontein.de/formulae/2PdJ1C7dVptN4T3KH.kcgsx26p_m.iEjRVAXRw.svgz" alt="T^{-1} A T = \Matrix{ \lambda_1 & * & \cdots & * \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots &
\ddots & * \\ 0 & \cdots & 0 & \lambda_n }" title="T^{-1} A T = \Matrix{ \lambda_1 & * & \cdots & * \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots &
\ddots & * \\ 0 & \cdots & 0 & \lambda_n }">
</div>
<p>
with <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="16" src="https://math.fontein.de/formulae/VYegolfuFS6jabfegkRnAip5FLJJWusyoUNz3A.svgz" alt="\lambda_1, \dots, \lambda_n \in \C" title="\lambda_1, \dots, \lambda_n \in \C"></span>. As the transcendence degree of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="15" src="https://math.fontein.de/formulae/yic83_sv9OvU5TDbFAcLOkDAihqEngiRZV4YIQ.svgz" alt="\Q" title="\Q"></span> is infinite, there exist elements <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="16" src="https://math.fontein.de/formulae/3i6e90gjT8u3rmw8mZ6EGJLr7CtT9CEXpcmpfw.svgz" alt="\mu_1, \dots, \mu_n \in \C" title="\mu_1, \dots, \mu_n \in \C"></span> such that for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/kgf70lvIqBRpoPoVKjSwiLnFcYYDjxjnNlkixQ.svgz" alt="j \in \N_{>0}" title="j \in \N_{>0}"></span>, the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="171" height="23" src="https://math.fontein.de/formulae/53znfIU_M2dgLF8Z1eufyvDZbnYGYze_SMihig.svgz" alt="\{ \lambda_i + \frac{1}{j} \mu_i \mid 1 \le i \le n \}" title="\{ \lambda_i + \frac{1}{j} \mu_i \mid 1 \le i \le n \}"></span> has exactly <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> elements. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="221" height="23" src="https://math.fontein.de/formulae/GYo1Ex.wxJQP1lLRtEkinCafIhXj7AKN3AMkHA.svgz" alt="A_j := A + \frac{1}{j} diag(\mu_1, \dots, \mu_n)" title="A_j := A + \frac{1}{j} diag(\mu_1, \dots, \mu_n)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/kgf70lvIqBRpoPoVKjSwiLnFcYYDjxjnNlkixQ.svgz" alt="j \in \N_{>0}" title="j \in \N_{>0}"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="17" src="https://math.fontein.de/formulae/1Lsk8_RGEPZpDckBx8h7AnkrLLHG10ME3DMxtQ.svgz" alt="A_j \to A" title="A_j \to A"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="15" src="https://math.fontein.de/formulae/9vyPMzjbkz2q4AwybM.Dhbv1AbrP2.riSMnJbg.svgz" alt="j \to \infty" title="j \to \infty"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="17" src="https://math.fontein.de/formulae/lYHZ0xg0W13_sOhhnP4k4vNY7Kq6MCk.ejQXsg.svgz" alt="A_j" title="A_j"></span> has <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> distinct eigenvalues for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="8" height="15" src="https://math.fontein.de/formulae/q4ijdcMcwrer_7oZd9BjA6ImUoXNyAWyR9W9Kg.svgz" alt="j" title="j"></span>, namely <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="23" src="https://math.fontein.de/formulae/D.Ox2qarlCZbmCCnLMgbr1DCM35xdIjoTKlllQ.svgz" alt="\lambda_1 +
\frac{1}{j} \mu_1, \dots, \lambda_n + \frac{1}{j} \mu_n" title="\lambda_1 +
\frac{1}{j} \mu_1, \dots, \lambda_n + \frac{1}{j} \mu_n"></span>. But this implies that <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="17" src="https://math.fontein.de/formulae/yYrna_jFJyhR1HdPxGf67QRWRpmygZpmY3CNsg.svgz" alt="A_j \in D" title="A_j \in D"></span>, whence we found a sequence in <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/.1gXmU_PRktx2Rs2BFoRNw.Q_4Ub6YJ34UcmGw.svgz" alt="D" title="D"></span> converging to <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now, we are able to conclude:
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
<a name="cayleyhamiltonoverC"></a>
Theorem (Cayley-Hamilton over the complex numbers).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="15" src="https://math.fontein.de/formulae/EeC2.bIXJd2Tfm7OiK.1Z8xFWDUjh09NZf5.ng.svgz" alt="A \in \C^{n \times n}" title="A \in \C^{n \times n}"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/hIyHTKO9mQ4kswRSfoSMpc1QZKR9c2jLqJFx6Q.svgz" alt="\chi_A(A) = 0" title="\chi_A(A) = 0"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="232" height="19" src="https://math.fontein.de/formulae/ECvnYPmWXb.qRGIZyEJYvE0CMQyzVEpVZnFsag.svgz" alt="S := \{ A \in \C^{n \times n} \mid \chi_A(A) = 0 \}" title="S := \{ A \in \C^{n \times n} \mid \chi_A(A) = 0 \}"></span>. Clearly, <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="15" src="https://math.fontein.de/formulae/6wALIjJoFxTmEr_QeoUP9u1aznHCGvIzvWKakA.svgz" alt="D \subseteq S" title="D \subseteq S"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/.1gXmU_PRktx2Rs2BFoRNw.Q_4Ub6YJ34UcmGw.svgz" alt="D" title="D"></span> is dense in <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="14" src="https://math.fontein.de/formulae/AGrdSL62ngoSFRvJP1JsZrPVccXNEtq4MbZs5Q.svgz" alt="\C^{n \times n}" title="\C^{n \times n}"></span> by the <a href="https://math.fontein.de/2009/05/04/a-topological-proof-of-the-cayley-hamilton-theorem-over-all-commutative-unitary-rings/#diagmatricesdenselemma">previous lemma</a>. Hence, it suffices to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is closed.
</p>
<p>
But note that the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="140" height="14" src="https://math.fontein.de/formulae/5szzeDxUMfcK0WPIGH0H8yAYwViyXhtiOTmAhA.svgz" alt="\Phi : \C^{n \times n} \to \C^{n \times n}" title="\Phi : \C^{n \times n} \to \C^{n \times n}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/AxryN21iFp8B47TrYl7UzudIL_.hpIXVZZ7weA.svgz" alt="A \mapsto \chi_A(A)" title="A \mapsto \chi_A(A)"></span> is defined by polynomials; hence, it is continuous. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="19" src="https://math.fontein.de/formulae/9qGTKKXLcvjdKUAnfLNDeiIABOH.ZhuVtZ6g1A.svgz" alt="S = \Phi^{-1}(\{ 0 \})" title="S = \Phi^{-1}(\{ 0 \})"></span> is the preimage of a closed set, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is closed itself.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
This completes the proof of the theorem:
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof (Cayley-Hamilton over commutative unitary rings).
</div>
<div class="theorem-content theorem-proof-content">
<p>
By the <a href="https://math.fontein.de/2009/05/04/a-topological-proof-of-the-cayley-hamilton-theorem-over-all-commutative-unitary-rings/#cayleyhamiltonreduction">first lemma</a>, it suffices to show the theorem over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>. But this is accomplished by the <a href="https://math.fontein.de/2009/05/04/a-topological-proof-of-the-cayley-hamilton-theorem-over-all-commutative-unitary-rings/#cayleyhamiltonoverC">previous theorem</a>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
</div>