Matrix Conjugation Series

The following are a series of posts that show what matrix conjugation is, and why it's so important. Basically, given a square matrix M and a non-singular matrix U, we can find a matrix that is similar by calculating:

U^-1 M U

Preliminaries

Matrix Trace, Determinants and Eigenvalues
Trace is used as the basis of several important functions on matrices in group theory and other aspects of linear algebra, but seriously lacks some simple-to-understand proofs on the Internet. Determinants come along for the ride.

Eigenmatrices, Decomposition and Conjugation
An introduction to how eigenmatrices extend the concept of eigenvectors/values and how this can be used to define two useful operations, diagonal decomposition and conjugation.

Uses of Decomposition and Conjugation

Matrix Conjugation Continued
This article shows how conjugation links different square matrices together, and how trace, determinant and eigenvalues are conserved under conjugation, important theories in several areas of mathematics.

Matrix Powers - Uses of Diagonalisation.
An article that shows how diagonal decomposition enables us to calculate large powers of a matrix. This serves as an introduction to:

Caley-Hamilton Theorem
This theory enables us to calculate any analytical (smooth, differentiable) function of a square matrix. It was the major reason I started writing this series of posts, and the series was essentially complete at this point.

Continuations:

These posts are posts that refer to the proofs and theories developed above:

Why Linear Differential Equations are Essentially First Order
This shows why linear differential equations are so easy to solve, and why they involve exponential solutions.