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 $\displaystyle A = P^{-1}DP$, where P is invertible and D is the diagonal matrix whose entries are the eigenvalues of A, then $\displaystyle A^n = P^{-1}D^nP$. 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 $\displaystyle P^{-1}D_0P$.
The eigenvalues of this matrix are $\displaystyle \lambda=\tfrac12$ and $\displaystyle \lambda=\tfrac18(5\pm\sqrt{105})$. Since $\displaystyle |\tfrac18(5+\sqrt{105})|>1$ 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.]