M = VDV

^{-1}This process is referred to as

__and has some useful consequences. In this post, we'll focus on taking the matrix powers of M.__

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

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.

M

= (VDV

= VDV

^{3}= MMM= (VDV

^{-1})(VDV^{-1})(VDV^{-1})= VDV

^{-1}VDV^{-1}VDV^{-1}It should be clear that the inner pairs of V and it's inverse cancel out to leave:

M

^{3}= V D^{3}V^{-1 }Likewise

M

^{100}= V D^{100}V^{-1}Since D is a

__matrix of M's eigenvalues, the value of D__

**diagonal**^{100}can be calculated by raising each value of D to the power of 100

__, saving a lot of time and effort.__

**elementwise**However, there is an even faster short-cut.

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