# Steady States

Printable View

• Nov 17th 2010, 06:51 PM
veronicak5678
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.
• Nov 17th 2010, 07:11 PM
pickslides
Quote:

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.
• Nov 17th 2010, 07:37 PM
veronicak5678
Could you please explain how you got that answer?
• Nov 18th 2010, 11:51 AM
pickslides
I used my calculator to find that matrix to a very high power. In this example either $A^{50}$ or $A^{100}$ will work.
• Nov 18th 2010, 03:18 PM
Ackbeet
The more complete method of solution involves diagonalizing the matrix. Let's say that for a matrix $A$ you could find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}.$ Then

$A^{2}=(PDP^{-1})(PDP^{-1})=PDDP^{-1}=PD^{2}P^{-1},$

$A^{3}=(PDP^{-1})(PDP^{-1})(PDP^{-1})=PDDDP^{-1}=PD^{3}P^{-1}.$

More generally,

$A^{k}=PD^{k}P^{-1}.$

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.