On the structure and geometry of the product singular value decomposition

Abstract

The product singular value decomposition is a factorization of two matrices, which can be considered as a generalization of the ordinary singular value decomposition, at the same level of generality as the quotient (generalized) singular value decomposition. A constructive proof of the product singular value decomposition is provided, which exploits the close relation with a symmetric eigenvalue problem. Several interesting properties are established. The structure and the nonuniqueness properties of the so-called contragredient transformation, which appears as one of the factors in the product singular value decomposition, are investigated in detail. Finally, a geometrical interpretation of the structure is provided in terms of principal angles between subspaces.