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.

Matrix Powers - Uses of Diagonal Decomposition and Conjugation

As mentioned in a previous post, nearly any square matrix M can be decomposed into a diagonal matrix D, using the eigenmatrix V:

M = VDV^-1

This process is referred to as diagonal decomposition and has some useful consequences. In this post, we'll focus on taking the matrix powers of M.

Say we want to calculate the 100th power of M:

M¹⁰⁰

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.