Mathematics

Questions tagged [hankel-matrices]

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.

53 questions
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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 $$\det \begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & (n+3)^{2}\\ (n+2)^{2} & (n+3)^{2} & (n+4)^{2} \end{bmatrix} = -8$$ Clearly,…
Shev
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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 & s_{n+1}\\ \vdots & \vdots &…
Vim
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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 &…
Keivan
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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
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Determinant of a Hankel matrix involving factorials

Consider the matrix $$M_n=\begin{bmatrix}1&\frac1{2!}&\frac1{3!}&\dotsb&\frac1{n!}\\ \frac1{2!}&\frac1{3!}&\frac1{4!}&\dotsb&\frac1{(n+1)!}\\ \frac1{3!}&\frac1{4!}&\frac1{5!}&\dotsb&\frac1{(n+2)!}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\…
Akiva Weinberger
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8
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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
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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 $\pm 1$ ). $$…
Apple
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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
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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
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5
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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} \end{bmatrix}…
Carl
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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
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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
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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
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4
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1 answer

A general Hankel matrix over ${\Bbb F}_q$ that is nonsingular

Let $${\bf M} = \begin{pmatrix} m_1 & m_2 &\cdots & m_\ell \\ m_2 & m_3 &\cdots & m_{\ell+1} \\ \vdots & \vdots &\ddots & \vdots \\ m_\ell & m_{\ell+1} &\cdots & m_{2\ell-1} \end{pmatrix} \in (\mathbb{F}_q)^{\ell \times \ell}$$ be a general Hankel…
Suneves
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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
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