Cauchy matrix
Encyclopedia
In mathematics
, a Cauchy matrix is an m×n matrix
with elements aij in the form
where
and 
are elements of a field
, and 
and 
are injective sequences (they do not contain repeated elements; elements are distinct).
The Hilbert matrix is a special case of the Cauchy matrix, where
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
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
(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
(Schechter 1959, Theorem 1)
where Ai(x) and Bi(x) are the Lagrange polynomials for
and 
, respectively. That is,
with
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
Here
denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Cauchy matrix is an m×n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with elements aij in the form
where
and are elements of a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and and are injective sequences (they do not contain repeated elements; elements are distinct).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 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
(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
(Schechter 1959, Theorem 1)where Ai(x) and Bi(x) are the Lagrange polynomials for
and , respectively. That is,with
Generalization
A matrix C is called Cauchy-like if it is of the formDefining 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
opsFloating pointIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
(e.g. the fast multipole methodFast Multipole MethodThe fast multipole method is a mathematical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem...
), - (pivoted) LU factorization with
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).