What are the limits as k -> infinity of the following:

[.4 .2; .6 .8]^k[1; 0]

[.4 .2; .6 .8]^k[0; 1]

[.4 .2; .6 .8]^k

I think the first two go to [1; 0] and the last goes to [0 0; 0 0], but am not sure.

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- November 17th 2010, 07:51 PMveronicak5678Steady States
What are the limits as k -> infinity of the following:

[.4 .2; .6 .8]^k[1; 0]

[.4 .2; .6 .8]^k[0; 1]

[.4 .2; .6 .8]^k

I think the first two go to [1; 0] and the last goes to [0 0; 0 0], but am not sure. - November 17th 2010, 08:11 PMpickslides
- November 17th 2010, 08:37 PMveronicak5678
Could you please explain how you got that answer?

- November 18th 2010, 12:51 PMpickslides
I used my calculator to find that matrix to a very high power. In this example either or will work.

- November 18th 2010, 04:18 PMAckbeet
The more complete method of solution involves diagonalizing the matrix. Let's say that for a matrix you could find an invertible matrix and a diagonal matrix such that Then

More generally,

But the kth power of a diagonal matrix is just the (diagonal) matrix with the diagonal elements of the original diagonal matrix raised to the kth power.

So you should be able to compute the limit exactly.