J-Orthogonal Matrices: Properties and Generation

A real, square matrix Q is J-orthogonal if QTJQ = J, where the signature matrix $J = \diag(\pm 1)$. 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. We present techniques and tools useful in the analysis, application, and construction of these matrices, giving a self-contained treatment that provides new insights. First, we define and explore the properties of the exchange operator, which maps J-orthogonal matrices to orthogonal matrices and vice versa. 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. We employ the decomposition to derive an algorithm for constructing random J-orthogonal matrices with specified norm and condition number. We also give a short proof of the fact that J-orthogonal matrices are optimally scaled under two-sided diagonal scalings. We introduce the indefinite polar decomposition and investigate two iterations for computing the J-orthogonal polar factor: a Newton iteration involving only matrix inversion and a Schulz iteration involving only matrix multiplication. We show that these iterations can be used to J-orthogonalize a matrix that is not too far from being J-orthogonal.