Cauchy matrix
In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an "m"×"n" matrix with elements "a" in the form where formula_2 and formula_3 are elements of a field formula_4, and formula_5 and formula_6 are injective sequences (they contain "distinct" elements). The Hilbert matrix is a special case of the Cauchy matrix, where 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 formula_5 and formula_6. If the sequences were not injective, the determinant would vanish, and tends to infinity if  some formula_2 tends to formula_3. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles: The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A = B = [b] is given by where "A"(x) and "B"(x) are the Lagrange polynomials for formula_5 and formula_6, respectively. That is, with Generalization. A matrix C is called Cauchy-like if it is of the form Defining X =diag(x), Y=diag(y), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation (with formula_20 for the Cauchy one). Hence Cauchy-like matrices have a common displacement structure, which can be exploited while working with the matrix. For example, there are known algorithms in literature for Here formula_24 denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).