Hankel matrix (or catalecticant matrix), is a square matrix in which each ascending skew-diagonal from left to right is constant. The Hilbert matrix is an example of a Hankel matrix.
Questions tagged [hankel-matrices]
51 questions
37
votes
3 answers
How to find the determinant of this $3 \times 3$ Hankel matrix?
Today, at my linear algebra exam, there was this question that I couldn't solve.
Prove that
$$\det \begin{bmatrix}
n^{2} & (n+1)^{2} &(n+2)^{2} \\
(n+1)^{2} &(n+2)^{2} & (n+3)^{2}\\
(n+2)^{2} & (n+3)^{2} & (n+4)^{2}
\end{bmatrix} =…
Shev
- 1,027
- 1
- 14
- 24
20
votes
2 answers
Any neat way to calculate this Vandermonde-like determinant?
Let $x_i,i\in\{1,\cdots,n\}$ be real numbers, and $s_k=x_1^k+\cdots+x_n^k$, I'm asked to calculate
$$
|S|:=
\begin{vmatrix}
s_0 & s_1 & s_2 & \cdots & s_{n-1}\\
s_1 & s_2 & s_3 & \cdots & s_n\\
s_2 & s_3 & s_4 & \cdots &…
Vim
- 13,905
15
votes
2 answers
Determinant of anti-circulant matrix
Find the determinant of the following matrix in the terms of $a_1,a_2,\dots,a_n$ explicitly.
$$\begin{bmatrix}
a_1 & a_2 & a_3 & \cdots & a_n\\
a_2 & a_3 & a_4 & \cdots & a_1\\
a_3 & a_4 & a_5 & \cdots & a_2\\
\vdots & \vdots & \vdots & \ddots &…
k1.M
- 5,577
12
votes
0 answers
A direct proof of the Vandermonde decomposition of a nonsingular Hankel matrix?
I have been doggedly searching for a direct proof of the following theorem:
Theorem 1: Let $H$ be a complex nonsingular $n\times n$ Hankel matrix. Then $H$ can be factorized $H = V^\top DV$ where $V$ is a complex $n\times n$ Vandermonde matrix and…
eepperly16
- 7,712
10
votes
4 answers
8
votes
2 answers
A curious Hankel determinant
Define the sequence
$$ a_{n} := \begin{cases} 1 & \text{if } n+1=2^k \text{ for some } k \\ 0 & \text{otherwise} \end{cases} $$
Computer experiments suggest that the determinant of the $(n+1) \times (n+1)$ Hankel matrix
$$ H_{n+1}:=\begin{pmatrix}
…
Johann Cigler
- 1,521
- 8
- 13
6
votes
2 answers
Determinants of certain Hankel matrices
I have a sequence of Hankel matrices $A_{n}$ (i.e. matrix whose entries on all anti-diagonals are the same), and its entries are either $1$ or $0$. I am trying to prove that their determinant are either $0$ or $\pm…
Apple
- 303
6
votes
2 answers
Linear algebra question: does it have a solution?
Given $k\in\mathbb{N}$, $p$ a prime number, $s = (s_1, s_2,..., s_{2k+1})\in \mathbb{M}_{(2k+1)*1}(\mathbb{F}_p)$, the Hankel matrix generated by $s$ is denoted as $H$ where
$$
H = \begin{pmatrix}
s_1 & s_2 & \cdots & s_{k+1} \\
s_2 & s_3 & \cdots &…
Youzhe Heng
- 61
5
votes
1 answer
Are these Hankel matrices positive semidefinite?
While working on a quantum information project, I encountered the following two Hankel matrices
$$ a_{i,j} = (i+j)!(2n-(i+j))! ,\qquad b_{i,j} = (i+j+1)!(2n-(i+j))! $$
where $0 \le i,j \le n$ and $!$ denotes the factorial. I would like to know if…
saikohage
- 53
- 4
5
votes
0 answers
Proving that the $n \times n$ Hilbert matrix is positive definite
Prove that the following matrix is positive definite. $$ A = \begin{bmatrix} 1 & \frac12 & \dots & \frac1n \\ \frac12 & \frac13 & \dots & \frac1{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ \frac1n & \frac1{n+1} & \dots & \frac1{2n-1}…
Carl
- 187
5
votes
2 answers
Rank preservation of Hankel matrix by adding constrained sample
Let some $x_i \in \mathbb{R}$ for every $i$ such that the Hankel matrix
$$H_0=\begin{bmatrix} x_0 & x_1 & x_2 & x_3\\
x_1 & x_2 & x_3 & x_4\\
x_2 & x_3 & x_4 & x_5
\end{bmatrix} $$
is full rank, with the rank equal to 3. Show that there exists some…
Betelgeuse
- 482
5
votes
2 answers
Hankel matrix of Catalan numbers
Recall that the $n$-th Catalan number $C_n=\frac{1}{n+1}{2n\choose n}$ counts the number of
paths connecting $(0, 0)$ to $(n, n)$ that travel along the grid of integer lattice points of
$R^2$ where each path moves up or right in one-unit steps and…
d.y
- 669
4
votes
1 answer
Putnam $2024$ — Determinant of a Hankel matrix
Problem $\text A6$ on the $2024$ William Lowell Putnam Mathematical Competition was as follows.
Let $c_0, c_1, c_2, \dots$ be the sequence defined so that $$ \frac{1 - 3x - \sqrt{1 - 14x + 9x^2}}{4} = \sum_{k=0}^{\infty} c_k x^k $$ for sufficiently…
Guy Fsone
- 25,237
4
votes
1 answer
A general Hankel matrix over ${\Bbb F}_q$ that is nonsingular
Let
$${\bf M} = \begin{pmatrix}
m_1 & m_2 &\cdots & m_\ell \\
m_2 & m_3 &\cdots & m_{\ell+1} \\
\vdots & \vdots &\ddots & \vdots \\
m_\ell & m_{\ell+1} &\cdots &…
Suneves
- 73
4
votes
1 answer
Product of Inverse Hankel Matrix
Consider $H_n$, the $n\times n$ Hankel matrix of the Catalan numbers starting from $2$:
$$H_n = \begin{bmatrix}
2 & 5 & 14 & 42 & 132\\
5 & 14 & 42 & 132 & 429\\
14 & 42 & 132 & 429 & 1430 & \cdots\\
42 & 132 & 429 & 1430 & 4862\\
132 & 429 & 1430 &…
yanjunk
- 219