## Archive for the ‘Linear Algebra’ Category

### Solving Certain Linear Systems over the Integers.

friday, june 17th, 2011

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.

### On a Certain Determinant.

friday, march 25th, 2011

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.

### Multiplicity of the Determinant.

wednesday, november 10th, 2010

This post shows a way to quickly show that the determinant is multiplicative without getting your hands dirty.

### Homomorphisms, Tensor Products and Certain Canonical Maps.

friday, january 29th, 2010

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.

### Diagonalizable Matrices.

friday, january 29th, 2010

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.

### About Base Changes and Tensor Products.

saturday, august 15th, 2009

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.

### A Note on the Jordan Decomposition.

tuesday, may 5th, 2009

We show some kind of universal property for the Jordan decomposition of an endomorphism of a finite dimensional vector space.

### A Topological Proof of the Cayley-Hamilton Theorem over all Commutative Unitary Rings.

monday, may 4th, 2009

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.