![]() Then we show how the exchange operator can be used to obtain a hyperbolic CS decomposition of a J-orthogonal matrix directly from the usual CS decomposition of an orthogonal matrix. ![]() ![]() First, we define and explore the properties of the exchange operator, which maps J-orthogonal matrices to orthogonal matrices and vice versa. We present techniques and tools useful in the analysis, application, and construction of these matrices, giving a self-contained treatment that provides new insights. J-orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products, including, in particular, the downdating of Cholesky factorizations. A real, square matrix Q is J-orthogonal if $Q^\ JQ=J$, where the signature matrix J = diag(☑).
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