  The singular value decomposition of a complex matrix is a fundamental concept
in linear algebra and has proved extremely useful in many subjects. It is less
clear what the situation is over a finite field. In this paper, we classify the
orbits of GU(m,q) x GU(n,q) on n by n matrices (which is the analog of the
singular value decomposition). The proof involves Kronecker's theory of pencils
and the Lang-Steinberg theorem for algebraic groups. Besides the motivation
mentioned above, this problem came up in a recent paper of Guralnick, Larsen
and Tiep where a concept of character level for the complex irreducible
characters of finite, general or special, linear and unitary groups was studied
and bounds on the number of orbits was needed. A consequence of this work
determines possible pairs of Jordan forms for nilpotent matrices of the form AB
where B is either the transpose of A or the conjugate transpose.
