Abstract
A simple and short proof of the optimality conditions in the John von Neumann trace inequality for singular values is shown. Possible generalizations and special cases are also considered.
Highlights
Let A and B be n x n real matrices, where a1 2 ... 2 an are the singular values of A, and f3I 2 ... 2 f3n the singular values of B
Tr denotes trace of a mat.rix and T its transpose. This inequality was first proved by John von Neumann [12] and has important applications in the study of certain equations of nonlinear elasticity
If we replace tr(ABT) by tr(AB) in the theorem we do not have a simultaneous singular decomposition for A and B, but we have from the proof of the theorem that AB and BA are symmetric
Summary
Let A and B be n x n real matrices, where a1 2 ... We will find that under this situation, both A and B have simultaneous singular decompositions. Its derivative with rcspect toe vanishes at e = o, and this implies that Cij = Cji· ABT is symmetric, and since tr(ABT) = tr(BT A), BT A is symmetric.
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