Cauchy matrix explained
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form
};\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n
where
and
are elements of a
field
, and
and
are
injective sequences (they contain
distinct elements).
Properties
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
The Hilbert matrix is a special case of the Cauchy matrix, where
Cauchy determinants
The determinant of a Cauchy matrix is clearly a rational fraction in the parameters
and
. If the sequences were not injective, the determinant would vanish, and tends to infinity if some
tends to
. 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
(xi-xj)(yj-yi)}\over
(xi-yj)}}
(Schechter 1959, eqn 4; Cauchy 1841, p. 154, eqn. 10).It is always nonzero, and thus all square Cauchy matrices are
invertible. The inverse
A−1 =
B = [b<sub>ij</sub>] is given by
(Schechter 1959, Theorem 1)where
Ai(x) and
Bi(x) are the
Lagrange polynomials for
and
, respectively. That is,
with
A(x)=
(x-xi) and B(x)=
(x-yi).
Generalization
A matrix C is called Cauchy-like if it is of the form
Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
(with
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
ops (e.g. the
fast multipole method),
ops (GKO algorithm), and thus linear system solving,
- approximated or unstable algorithms for linear system solving in
.Here
denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
See also
References
