Concept

Cauchy matrix

Summary
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form : a_{ij}={\frac{1}{x_i-y_j}};\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n where x_i and y_j are elements of a field \mathcal{F}, and (x_i) and (y_j) are injective sequences (they contain distinct elements). The Hilbert matrix is a special case of the Cauchy matrix, where :x_i-y_j = i+j-1. ; Every submatrix of a Cauchy matrix is itself a Cauchy matrix. Cauchy determinants The determinant of a Cauchy matrix is clearly a rational fraction in the parameters (x_i) and (y_j). If the sequences were not injective, the determinant would vanish, and tends to infinity if some x_i tends to y_j. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles: The determinant of
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