Thread: Steady 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.

Originally Posted by veronicak5678

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

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

[0.25 0.25;0.75 0.75]

This should help you with the first 2.

Could you please explain how you got that answer?

I used my calculator to find that matrix to a very high power. In this example either or will work.

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.