On the singular value decomposition over finite fields and orbits of GU x GU

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.