Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [cauchy-matrices]

A Cauchy matrix is one with entries $a_{ij} = \frac{1}{x_i \pm y_j}$.

A Cauchy matrix, is an $m\times n$ matrix with elements $a_{ij}$ in the form

$$a_{ij}=\frac{1}{x_i-y_j};\quad x_i-y_j\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n$$

where $x_i$ and $y_j$ are elements of a field $\mathbb{F}$, and $(x_i)$ and $(y_j)$ are injective sequences.

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.

19 questions
16
votes
1 answer

Connection between the spectra of a family of matrices and a modelization of particles' scattering?

In the excellent book "Numerical Computing with MATLAB" by Cleve B. Moler (SIAM 2004), [Moler is the "father" of Matlab], one finds, on pages 298-299, the following graphical representation (fig. 10.12 ; I have reconstructed it with minor changes ;…
Jean Marie
  • 88,997
5
votes
4 answers

Proving a symmetric Cauchy matrix is positive semidefinite

Let $x_1, x_2,\dots, x_n$ be positive real numbers. Let $A$ be the $n\times n$ matrix whose $i,j^\text{th}$ entry is $$a_{ij}=\frac{1}{x_i+x_j}.$$ This is a Cauchy matrix. I am trying to show that this matrix is positive semi-definite. I have been…
R_D
  • 7,520
4
votes
1 answer

History of the Cauchy matrix

Cauchy matrix $C$ is defined by $$C_{i,j}=\frac{1}{a_i + b_j}$$ where $a_i$ and $b_j$ are any numbers so long $a_i + b_j \neq 0$. Why did Cauchy introduce this matrix? Did he use it in the context of another problem or application?
Maesumi
  • 3,722
3
votes
0 answers

Express Lagrange polynomial in term of Cauchy matrix

Given 2n distinct real numers $s_1,s_2, \dots, s_n$ and $t_1, t_2, \dots,t_n$ define the $n \times n$ Cauchy matrix $C = C(t,s)$ by $C_{ij} = \frac{1}{t_i - s_j}$. Express the Lagrange interpolation formula: $$p(t_i) =…
Alexander
  • 1,281
2
votes
1 answer

Complexity of multiplying Cauchy matrix by a vector

I want to find the matrix-vector product: $b=C \times a$ where $b=[b_1, ..., b_n]$ is the unknown vector, $a=[a_1, ..., a_n]$ is the known vector, and C is the coefficients matrix. The complexity to calculate b is $O(n^2)$ Assuming that the C matrix…
sam
2
votes
1 answer

Matrices that are simultaneously Cauchy matrices and Toeplitz ones

The article https://www.sciencedirect.com/science/article/pii/002437959190321M defines "Cauchy-Toeplitz matrices" those matrices that are simultaneously Cauchy matrices and Toeplitz ones. Then the author writes that: It is easy to check that an…
user222167
  • 23
2
votes
1 answer

Determinant of $(1/(a_i+a_j))$

Let $a_1,\ldots,a_p$ be positive real numbers. Let $A = (a_{ij})_{p\times p}$ where $a_{ij} = 1/(a_i+a_j)$. How to prove that $$\det A = 2^{-p}\prod_{j=1}^p \frac{1}{a_j}\prod_{1\leq j
Stephen
  • 866
2
votes
1 answer

Please recommend some material about Cauchy matrices to me

Recently I am trying to compute some variation of Cauchy determinant like this $$\det\left(\frac{1}{(x_i+y_j)^k}\right)$$ I have just learned from Google that Borchardt computed the above determinant when…
Laura
  • 4,769
2
votes
1 answer

Proof for the determinant of a Cauchy matrix

I want to proof the formula for the determinant of a Cauchy Matrix without recurring to matrix manipulation, but by directly applying the definition of the determinant. That is, given two sequences of numbers of length n, $x_1,...,x_n$ and…
Ezrael
  • 35
2
votes
1 answer

How to generate a Cauchy matrix for Reed-Solomon Coding by hand?

I am trying to understand Cauchy Reed-Solomon Coding based on the paper Optimizing Cauchy Reed-Solomon Codes for Fault-Tolerant Storage Applications. I am referring to the following section: An $m×n$ Cauchy matrix is defined as follows. Let $X =…
1awuesterose
  • 133
1
vote
0 answers

Rank of any submatrix of a Cauchy matrix

Here said that any $n \times n$ submatrix of a $m \times n$ cauchy matrix has full rank, where $m > n$. Question: Is it possible to generalize the statement into non-sqaure submatrix? That is, any $a \times b$ submatrix of a cauchy matrix has full…
Idonknow
  • 16,493
1
vote
0 answers

Determinant of sparse Hilbert matrix

It is known that the determinant of the Hilbert matrix of dimension $N$ with elements $$ H^N_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N $$ namely of the form \begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\ \frac{1}{2} &…
knuth
  • 31
1
vote
0 answers

Determinant of Hilbert-like matrix

It is known that the determinant of the Hilbert matrix with elements $$ H_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N $$ decreases exponentially to zero. It can be proven that the Hilbert-like matrix with elements $$ H_{tr} = \int_0^\infty d\omega \,…
knuth
  • 31
1
vote
2 answers

Verifying a polynomial is positive on the half-line defined by $z \geq 1$

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the polynomials that I generate using my rules must…
Stefan Smith
  • 8,322
1
vote
1 answer

Finding the determinant of a Cauchy matrix

Found this nice matrix: $$\begin{pmatrix} \dfrac{1}{a_1+b_1} & \dfrac{1}{a_2+b_1} & \cdots & \dfrac{1}{a_n+b_1} \\ \dfrac{1}{a_2+b_1} & \dfrac{1}{a_2+b_2} & \cdots & \dfrac{1}{a_n+b_2}\\ \vdots & \vdots &\ddots & \vdots \\ \dfrac{1}{a_n + b_1} &…
cgo
  • 1,970
1
2