Archive for the ‘Linear Algebra’ Category
We present a (well-known) method to compute a solution to the linear system Ax=b over the integers, when it is known that the determinant of A is non-zero and that a solution with integral coefficients exists. We also provide a running time analysis.
In this post, I show how to explicitly compute a determinant. This determinant allows me to write down a closest solution in the 2-norm to a certain unsolvable linear system.
This post shows a way to quickly show that the determinant is multiplicative without getting your hands dirty.
A standard topic in linear algebra is the dual space of a vector space, as well as the canonical embedding of a vector space in its double dual. Moreover, transposition of homomorphisms in terms of dual spaces is rather well known. Something less known is that one has a canonical map from the dual of V tensored with W to the space of homomorphisms from V to W. In this abstract nonsense post, we describe these canonical maps, their interplay, and try to determine their images.
We consider the property of an n times n matrix of being diagonalizable. Is this property open in the standard topology, or the Zariski topology? The emphasis lies on the real and complex numbers, as well as on arbitrary algebraically closed fields.
In Linear Algebra, one often has the problem that one wants to talk about complex eigenvalues of objects defined over the reals. If the object is a matrix, it is clear what that means. But what if the object is an endomorphism of a non-canonical real vectorspace? This question is strongly related an important use of tensor products, namely base changes.
Functional Calculus in Linear Algebra, the Jordan Decomposition Reloaded and Cayley-Hamilton’s Theorem.thursday, august 13th, 2009
We explain the aims of functional calculus and specialize to polynomials evaluated at endomorphisms. We reconsider the Jordan decomposition and prove it with more generality. Then, we discuss Taylor expansion in the nilpotent part for endomorphisms with separable minimal polynomials, and prove Cayley-Hamilton again for arbitrary fields.
We show some kind of universal property for the Jordan decomposition of an endomorphism of a finite dimensional vector space.
We want to give a proof of the Cayley-Hamilton Theorem for all commutative rings with unity, which first reduces to the case of the field of complex numbers and then applies a topological argument.