A Cauchy matrix has entries with nonzero denominators. If are distinct, it is positive definite: is the Gram matrix of linearly independent exponential functions. This form occurs in finite-rank inverse scattering transform reconstructions.
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A Cauchy matrix is a type of structured matrix that is defined by its elements as follows: If \( a_1, a_2, \ldots, a_m \) and \( b_1, b_2, \ldots, b_n \) are two sequences of distinct numbers, the Cauchy matrix \( C \) formed from these sequences is an \( m \times n \) matrix defined by: \[ C_{ij} = \frac{1