Matrix Factoring

Matrixes can be factored into nice forms, especially forms like this:

 A = B C B^(-1).

This is nice, because

 A^n = B C^n B^(-1)

Well, this is nice if it's easier to compute 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,

 [ 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?

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So this is why the EigenValues correspond to (1 + interest rate).

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

Look what happens when we take C^n :

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

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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 ]

B^(-1) is the MatrixInverse of B.

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CategoryMath