Explaining convergence using special decomposition (linear algebra)?
There is a matrix P = [.2 .8; .8 .2]
We are told to (using MATLAB) calculate P^n, where the largest n is 20. From this, the matrix P^n looks to approach [.5 .5; .5 .5]
The question is to explain this convergence using the special decomposition of P. There is a formula in the textbook that I am pretty sure I am using wrong, but this question is not meant to be answered using calculations, but by explaining it using words.
Any help would be appreciated.
Update:There was a typo on the assignment and it should say spectral decomposition, not special
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You can prove by induction that
P^n = ½ ( [1, 1; 1, 1] + (-3/5)^n [1, -1; -1, 1] ).
It would be more suggestive to write
P^n = ½ ( (1)^n [1, 1; 1, 1] + (-3/5)^n [1, -1; -1, 1] ),
since the two eigenvalues of P are 1 and -3/5.
What happens to eigenvalues of magnitude <1,
=1, and >1 when n→∞?
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