Consider two matrices and which in the absence of error have the same Schur vectors; i.e., there is a such that and are both block upper triangular where is the set of by orthogonal matrices. Now suppose that and are somewhat noisy from measurement errors or some other kind of lossy filtering. In that case the that upper triangularizes might not upper triangularize as well. How does one find the best ?
This is a problem that was presented to us by Schilders , who phrased it as a least squares minimization of , where is a mask returning the block lower triangular part of , where is broken up into blocks.
For this problem the differential is a bit tricky and its derivation instructive:
With second derivatives given by