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

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