In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an m×n matrix with elements aij in the form
a i j = 1 x i − y j ; x i − y j ≠ 0 , 1 ≤ i ≤ m , 1 ≤ j ≤ n where x i and y j are elements of a field 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.
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 a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as
det A = ∏ i = 2 n ∏ j = 1 i − 1 ( x i − x j ) ( y j − y i ) ∏ i = 1 n ∏ j = 1 n ( x i − y j ) (Schechter 1959, eqn 4).
It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = [bij] is given by
b i j = ( x j − y i ) A j ( y i ) B i ( x j ) (Schechter 1959, Theorem 1)
where Ai(x) and Bi(x) are the Lagrange polynomials for ( x i ) and ( y j ) , respectively. That is,
A i ( x ) = A ( x ) A ′ ( x i ) ( x − x i ) and B i ( x ) = B ( x ) B ′ ( y i ) ( x − y i ) , with
A ( x ) = ∏ i = 1 n ( x − x i ) and B ( x ) = ∏ i = 1 n ( x − y i ) . A matrix C is called Cauchy-like if it is of the form
C i j = r i s j x i − y j . Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
X C − C Y = r s T (with r = s = ( 1 , 1 , … , 1 ) 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
approximate Cauchy matrix-vector multiplication with O ( n log n ) ops (e.g. the fast multipole method),(pivoted) LU factorization with O ( n 2 ) ops (GKO algorithm), and thus linear system solving,approximated or unstable algorithms for linear system solving in O ( n log 2 n ) .Here n denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
