>> Under certain conditions,[4] the matrices Ak converge to a triangular matrix, the Schur form of A.
> So am I to understand that for A positive semi-definite, the A_{k} converges to a diagonal matrix?
Yes. Each iteration of QR (and LR) turns a PSD matrix into another PSD matrix. A PSD matrix in Schur form is necessarily a diagonal matrix.
>> The basic QR algorithm can be visualized in the case where A is a positive-definite symmetric matrix.
> It sounds like you can visualize the iterates A_{k} no matter what. Is the problem that there isn't a fixed point when the ellipsoid is axis-aligned?
My visualization technique assumes that the matrix is PSD. In the PSD case, there is a one-to-one correspondence between ellipsoids and PSD matrices. I don't know what happens if you apply it to non-PSD matrices, but the one-to-one correspondence gets lost.
> So am I to understand that for A positive semi-definite, the A_{k} converges to a diagonal matrix?
Yes. Each iteration of QR (and LR) turns a PSD matrix into another PSD matrix. A PSD matrix in Schur form is necessarily a diagonal matrix.
>> The basic QR algorithm can be visualized in the case where A is a positive-definite symmetric matrix.
> It sounds like you can visualize the iterates A_{k} no matter what. Is the problem that there isn't a fixed point when the ellipsoid is axis-aligned?
My visualization technique assumes that the matrix is PSD. In the PSD case, there is a one-to-one correspondence between ellipsoids and PSD matrices. I don't know what happens if you apply it to non-PSD matrices, but the one-to-one correspondence gets lost.