# Convergence of matrix equations

1. Oct 12, 2004

### gogetagritz

I am pretty rusty/unknowledgable when it comes to linear algebra, so when I was given the problem.

Find the limit as n->infinity of A^n * b

where a is a 2x2 matrix and b is a 2x1 vector, I scratched my head.

A fellow student told me about the spectral radius being the largest eigen value of A, and if it is less than one then the equation converges to zero. However my eigenvalues are not less than one. So what methods are there for determing an actual x,y that this this system converges to?

2. Oct 13, 2004

### matt grime

since A is 2x2 it satisfies a second order polynomial, the characteristic equation.

X^2+ uX + v=0 for some u,v in R

So, A^n = -uA^{n-1}-vA^{n-2}

for n greater than 2.

if A^nb converges to anything at all can you see how to find what it converges to now?