Cayley-Hamilton Theorem

My final post (so far) on matrix conjugation is the most useful of matrix theories and one of my personal favourites.

"In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation."

(http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem )

This means that once you have the characteristic polynomial of a matrix M (for a 3x3 matrix in this example):

P(x) = a₀ + a₁x + a₂x² + x³= 0

then replacing x with M:

P(M) = a₀I + a₁M + a₂M² + M³= 0

will also be true.

Notice that we've replaced the constant term a₀ with a₀I, guaranteeing that this will give us a square matrix.

Matrix Conjugation Continued.

As mentioned in an earlier post, nearly every square matrix is linked to a diagonal matrix consisting of that matrix's eigenvalues.

M = VDV^-1 and V^-1MV = D

This pre/post multiplication by a matrix and it's inverse is referred to as conjugation and interesting things happen if we use any non-singular square matrix U for this operation.

Eigenmatrices, Decompositions and Conjugation(Matrices)

 The definition of eigenvalues and eigenvectors states that for a matrix M, we can find pairs of vectors (v) and scalars (λ) that satisfy the following rule:

Mv = vλ

We can extend this concept to an eigenmatrix by combining all the n eigenvectors into an n x n matrix that we will call V, and replacing λ with a diagonal matrix D.

Matrix Trace, Determinants and Eigenvalues

This is the start of a quick series of posts showing some powerful properties (and proofs) of matrix eigenvalues. It is assumed that the reader can calculate eigenvectors/values.

However, the series refers to some basic algebra and matrix properties that, while used everywhere, usually don't have accessible or comprehensive  proofs available on the Internet (You can find them in good linear algebra textbooks). It is also a good place to show some neat tricks for sagemath.
There are better and more comprehensive proofs out there, but these are a good start.

We will start today with matrix traces and determinants, and their relationship to matrix eigenvalues. Basically we will be proving that for any n x n matrix, the eigenvalues add up to the sum of the diagonal elements, and multiply up to the determinant of the matrix.