Jordan Decomposition to Schur Decomposition
Let A be a complex or real square matrix. Suppose we have a Jordan decomposition A = , where X is non-singular and J is upper bidiagonal. Show how you can obtain a Schur Decomposition from a Jordan Decomposition.
Schur Decomposition: A = QTQ*, where Q is unitary/orthogonal and T is upper triangular with the eigenvalues of A on the diagonal.
I'm really not sure at all what to do. Because Q is orthogonal, Q*= . I'm not sure if that plays in somehow.
I've been trying to use SVDs or QR decompositions of X or J to get there, but I've had no luck. Does anyone have any suggestions? Thank you so much.