In
mathematics, a Cauchy matrix, named after
Augustin-Louis Cauchy, is an m×n
matrix with elements aij in the form
![{\displaystyle a_{ij}={\frac {1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0b9e5c440d839c4a53ffe75386d844bbb576bd)
where
and
are elements of a
field
, and
and
are
injective sequences (they contain distinct elements).
The
Hilbert matrix is a special case of the Cauchy matrix, where
![{\displaystyle x_{i}-y_{j}=i+j-1.\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f54fb52ee65c43814e1450a3ac5b84ffd1268601)
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; Cauchy 1841, p. 154, eqn. 10).
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,
![{\displaystyle A_{i}(x)={\frac {A(x)}{A^{\prime }(x_{i})(x-x_{i})}}\quad {\text{and}}\quad B_{i}(x)={\frac {B(x)}{B^{\prime }(y_{i})(x-y_{i})}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f76d84b1886b262bab600c2da79d5b6dca7ee2)
with
![{\displaystyle A(x)=\prod _{i=1}^{n}(x-x_{i})\quad {\text{and}}\quad B(x)=\prod _{i=1}^{n}(x-y_{i}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cef5d7e6855ef89a9973507c482c9d49e755ec7)
Generalization
A matrix C is called Cauchy-like if it is of the form
![{\displaystyle C_{ij}={\frac {r_{i}s_{j}}{x_{i}-y_{j}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb626b7a40be5cfb572745031781843d63564fbd)
Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the
displacement equation
![{\displaystyle \mathbf {XC} -\mathbf {CY} =rs^{\mathrm {T} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/242fee9990f325b51a7d30d7900c9f8cec2e3215)
(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),
- (
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).
See also
References