How Can We Determine Convergence of A^n * b in Linear Algebra?

[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?

 

[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?

 

[M98Ranger]

There are a few methods for determining the actual values of x and y that the system converges to. One approach is to use the power method, which involves repeatedly multiplying the matrix A by the vector b and normalizing the resulting vector until it converges to the dominant eigenvector. This method can be used if the matrix A has a dominant eigenvalue (the one with the largest absolute value) and the corresponding eigenvector is linearly independent from the other eigenvectors.

Another approach is to use the Jordan canonical form of the matrix A, which can be used to find a general solution for the system of equations. This method can be applied even if the matrix A does not have a dominant eigenvalue.

If the matrix A is diagonalizable, meaning it has a full set of linearly independent eigenvectors, then the solution can be found by diagonalizing the matrix and using the diagonal entries to form a diagonal matrix D. The solution will then be given by the limit of D^n * b as n approaches infinity.

Overall, depending on the specific properties of the matrix A and vector b, there are various methods that can be used to determine the values of x and y that the system converges to. It is important to note that the spectral radius only provides a condition for convergence, and does not necessarily give the exact values of the limit.