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

We could do this by multiplying out M by matrix multiplication 100 times, but decomposition provides us with a handy shortcut. We'll start with the 3rd power of M to make the process clearer.

M³ = MMM
= (VDV^-1)(VDV^-1)(VDV^-1)
= VDV^-1VDV^-1VDV^-1

It should be clear that the inner pairs of V and it's inverse cancel out to leave:

M³ = V D³ V^-1

Likewise

M¹⁰⁰ = V D¹⁰⁰ V^-1

Since D is a diagonal matrix of M's eigenvalues, the value of D¹⁰⁰ can be calculated by raising each value of D to the power of 100 elementwise, saving a lot of time and effort.

However, there is an even faster short-cut.