Matrixes can be factored into nice forms, especially forms like this:
**C**^n than it is to compute **A**^n! The nicest case is when the matrix has dim(**A**) unique EigenValues - in this case you can factor the matrix so that **C** has the EigenValues on the diagonal and zeros everywhere else, and the columns of **B** are the EigenVectors for the EigenValues !
For example,

*So this is why the EigenValues correspond to (1 + interest rate).*
*Look what happens when we take C^n :*
*The interest just keeps compounding...*
*The same rates of compounding happen within the original matrix, but are harder to tease out.*
NetPresentValue calculations use this feature of EigenValues.

So, for the example on the EigenValue page, we get:**B**^(-1) is the MatrixInverse of **B**.

CategoryMath

This is nice, becauseA=BCB^(-1).

Well, this is nice if it's easier to computeA^n =BC^nB^(-1)

[ 0 1 ] [ 1 0 ]Can be factored as

[ 1 -1 ] [ 1 0 ] [ 1/2 1/2 ] [ 1 1 ] [ 0 -1 ] [ -1/2 1/2 ]Cool, huh?

[ p 0 0 ]C= [ 0 q 0 ] [ 0 0 r ]

[ p^n 0 0 ]C^n = [ 0 q^n 0 ] [ 0 0 r^n ]

So, for the example on the EigenValue page, we get:

[ 0.97 0.35 ] [ 0.795 8.805 ] [ 1.06025 0 ] [ 1.2223 1.3789 ] [ ] ^n = [ ] * [ ] ^n * [ ] [ 0.08 0.75 ] [ 0.205 -7.805 ] [ 0 0.65975 ] [ 0.0321 -0.1245 ] [ 0.97 0.35 ]A= [ ] [ 0.08 0.75 ] [ 0.795 8.805 ]B= [ ] [ 0.205 -7.805 ] [ 1.06025 0 ]C= [ ] [ 0 0.65975 ] [ 0.9744 1.09925 ]B^(-1) = [ ] [ 0.0256 -0.09925 ]

CategoryMath

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