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CauchyMatrix

CauchyMatrix[x, y] is the matrix with entry (i, j) equal to 1/(x_i + y_j). Wolfram keeps the generating vectors in a structured array:

$ wo 'CauchyMatrix[{1, 2, 3}, {1, 5, 6}]'
CauchyMatrix[StructuredArray`StructuredData[{3, 3}, {{1, 2, 3}, {1, 5, 6}}]]

Normal expands it to the explicit matrix, which may be rectangular:

$ wo 'Normal[CauchyMatrix[{1, 2, 3}, {1, 5}]]'
{{1/2, 1/6}, {1/3, 1/7}, {1/4, 1/8}}

CauchyMatrix[x] is CauchyMatrix[x, x], and TargetStructure -> "Dense" returns the matrix directly:

$ wo 'CauchyMatrix[{1, 2, 3}, TargetStructure -> "Dense"]'
{{1/2, 1/3, 1/4}, {1/3, 1/4, 1/5}, {1/4, 1/5, 1/6}}

Given an explicit Cauchy matrix, the generating vectors are recovered, normalized so that y_1 is 0:

$ wo 'CauchyMatrix[{{1/2, 1/3}, {1/3, 1/4}}]'
CauchyMatrix[StructuredArray`StructuredData[{2, 2}, {{2, 3}, {0, 1}}]]