Is anyone familiar with how to do the following:
If given a matrix, A, find the limit of A^n as n approaches infinity.
I'm sorry that I don't have a specific example; can anyone explain it in a general case?
Thanks
Is anyone familiar with how to do the following:
If given a matrix, A, find the limit of A^n as n approaches infinity.
I'm sorry that I don't have a specific example; can anyone explain it in a general case?
Thanks
The usual technique for doing this (in cases where the limit exists) is to diagonalise the matrix. If , where P is invertible and D is the diagonal matrix whose entries are the eigenvalues of A, then . The matrix D^n is easy to compute: it is diagonal, and its entries are those of D raised to the power n. The limit matrix will only exist if all the eigenvalues are either 1 or have absolute value less than 1. In that case, D^n will converge to a limit matrix D_0, and A^n will converge to .
The eigenvalues of this matrix are and . Since it follows that the limit of A^n as n→∞ does not exist.
[Did you perhaps intend the 2 on the bottom row of the matrix to be a 0? The usual setting in which the limit of A^n is used is when the matrix A is stochastic, in which case I would expect to see a 0 where that 2 is.]