An analog of the singular value decomposition for complex orthogonal equivalence
Dipa Choudhury, Roger A. Horn · Linear and Multilinear Algebra · 1987
We show that a complex m-by-n matrix A can be factored as A=PΛQT , where P and Q are complex orthogonal matrices and Λ=[λ ij ] is an m-by-n generalized diagonal matrix (λ ij =0 if i ≠ j), if and only if ATA is diagonalizable and rank A = rank ATA. If is a family of m-by-n complex matrices, we show that there is a simultaneous factorization of this type if and only if each Ai has such a factorization and the products are symmetric for all These results are analogues (for complex orthogonal equivalence) of known theorems for unitary equivalence.