Let matrix A be symmetric with eigenvalues in the interval $[10,20]$.
1) in what interval are the eigenvalues of $A^2$ located?
2)in what interval are the eigenvalues of $P(A)$ where $P(x)$ is a polynomial located?
I am not too sure since i am not too familiar with the transformation of eigenvalues and so far I have been praciticing on iterative methods for solving matrices. But I suppose this has nothing to do with iterative methods? I know that if λ is an eigenvalue of a matrix A with corresponding eigenvector x, then λ2 is an eigenvalue of A2 with corresponding eigenvector x so for A the range is [100,400], i believe. What about the second one?
trivial question: is there any way to do this using iterative method?
If $A$ is a real symmtric matrix, then it is diagonalisable and its eigenvalues are real. So, $A$ is similar to a diagonal matrix $D$ and entries of the main diagonal of $D$ are the eigenvalues of $A$. So, $A^2$ is similar to $D^2$ and therefore the eigenvalues of $A^2$ belong to $[10^2,20^2]$.