Spectrum and trace invariance criterion and its statistical applications (Q750561)

Let \(A,B\in {\mathbb{C}}^{m\times n}\) be rectangular matrices. The authors give a simple condition involving the ranges of A, B and their conjugates which is necessary and sufficient for the set of nonzero eigenvalues of the product \(B^-A\) to be invariant including multiplicities with respect to the choice of a generalized inverse \(B^-\). It turns out that the condition is equivalent to \(tr B^-A\) being invariant. The result is then applied to investigate properties of canonical correlations in the context of the general Gauss-Markov model.