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A matrix $K\in \mathbb{R}^{n\times n}$ is Cauchy-like if its entries have the form
$$ K_{ij} = \frac{a_ib_j}{x_i-y_j} , \qquad i,j = 1,\ldots,n , $$ where $x_i,y_j$ for $i,j = 1,\ldots,n$ are pairwise distinct real numbers. Besides their pervasive occurrence in computations with rational functions, Cauchy-like matrices play an important role in deriving algebraic and computational properties of many displacement-structured matrix families and occur as fundamental blocks (together with trigonometric transforms) in decomposition formulas and fast solvers for, e.g., Toeplitz, Hankel, and related matrices.
This contribution provides a complete description of orthogonal Cauchy-like matrices [1]. Interest in these matrices stems from the paper [2], where they are needed to design all-pass filters for signal processing, and a novel characterization of Cauchy matrices as transition matrices between eigenbases of particular matrix pairs [3]. We illustrate their relationships with secular equations, the diagonalization of symmetric quasi-separable matrices and the construction of orthogonal rational functions with free poles. Moreover, we characterize matrix algebras that are simultaneously diagonalized by orthogonal Cauchy-like matrices.
[1] D. Fasino, Orthogonal Cauchy-like matrices, Numerical Algorithms, 92 (2023), 619--637.
[2] S. J. Schlecht, Allpass feedback delay networks, IEEE Trans. Signal Process., 69 (2021), 1028--1038.
[3] A. G. Lynch, Cauchy pairs and Cauchy matrices, Linear Algebra Appl., 471 (2015), 320--345.
