Skip to main content

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](https://math.stackexchange.com/questions/tagged/hilbert-matrices) is an example of a Hankel matrix.

Filter by
Sorted by
Tagged with
6 votes
2 answers
269 views

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 ...
Youzhe Heng's user avatar
1 vote
0 answers
27 views

Does every positive semidefinite hankel matrix obeys one Vandermonde decomposition?

I'm reading the paper(I can't find one arXiv version of this paper...) and suspect the correctness of one theorem inside. A hankel matrix $H$ is a square matrix in which each ascending skew-diagonal ...
narip's user avatar
  • 67
0 votes
0 answers
19 views

Hankel Singular Values for diagonal state-space model

Let (A,B,C) be a diagonal and stable discrete-time LTI state space model: \begin{align*}x(k+1)&=Ax(k)+Bu(k),\\ y&= Cx(k)\end{align*} with A being a diagonal matrix: \begin{bmatrix} \lambda_1 &...
Time Pass's user avatar
1 vote
0 answers
49 views

Decomposition of a matrix into observability and controllability matrices

$\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}$ I have a matrix $\boldsymbol{Q} \in \mathbb{R}^{M \times M}$ in ...
Neuling's user avatar
  • 57
0 votes
0 answers
36 views

Vandermonde matrix and its relation with Hankel matrix

Suppose I have a system of equation $Y = VX$, where $V$ is a tall Vandermonde and full-rank: $$V \triangleq\left(\begin{array}{ccccc} 1 & \lambda_1 & \lambda_1^2 & \lambda_1^3 & \ldots ...
yes's user avatar
  • 888
2 votes
2 answers
105 views

Hankel Operator is compact

I am currently working on compact operators. I am trying to solve the following exercise Problem Let $(a_j)_{j \in \mathbb{N}}$ be a sequence of complex numbers in $\ell_1$, i.e. $\sum_j |a_j| < \...
liamsi Meean's user avatar
0 votes
1 answer
72 views

Is the following Hankel matrix positive definite?

The matrix whose positive definiteness I am investigating is the following one $$ H_k(p)=\begin{bmatrix}1 & \frac{1}{p+1} & \frac{1}{p+2} & \frac{1}{p+3} & ... & \frac{1}{p+k-1} \...
Ann's user avatar
  • 9
1 vote
0 answers
74 views

The sign of a determinant?

Let $$A = [a_{ij}] = \begin{bmatrix} 1& 1/2! & 1/3!\\ 1/2! & 1/3!& 1/4!\\ 1/3!& 1/4! & 1/5!\end{bmatrix}.$$ Let $A^{\circ r}=[a_{ij}^r]$ for $r>0$. We need to prove that $\...
VSP's user avatar
  • 331
5 votes
1 answer
156 views

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 $!$ ...
saikohage's user avatar
2 votes
1 answer
121 views

Integral 6.717 of Gradshteyn and Ryzhik: $\int_{-\infty}^\infty \frac{\sin(a(x+\beta))}{x^\nu(x+\beta)} J_{\nu+2n}(x) \,dx$

I'm looking for a proof of the Bessel integral identity 6.717 from Gradshteyn and Ryzhik's Tables of Integrals, Series and Products, namely $$ \int_{-\infty}^\infty \frac{\sin(a(x+\beta))}{x^\nu(x+\...
heiner's user avatar
  • 1,098
5 votes
0 answers
173 views

Proving that the $n \times n$ Hilbert matrix is positive definite [duplicate]

Prove that the following matrix is positive definite. $$ A = \begin{bmatrix} 1 & \frac12 & \dots & \frac1n \\ \frac12 & \frac13 & \dots & \frac1{n+1} \\ \vdots & \vdots &...
Carl's user avatar
  • 189
3 votes
0 answers
186 views

A determinant similar to the Vandermonde'a determinant. [closed]

$$ \det\begin{bmatrix} 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&\vdots&\ddots&\...
Ashtart's user avatar
  • 123
4 votes
1 answer
191 views

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 &...
yanjunk's user avatar
  • 219
3 votes
1 answer
1k views

Why Hankel matrices?

Given a finite sequence $h = (h_0,h_1,h_2,\ldots,h_{2n-2})$, one can form the Hankel matrix $$ H = \begin{bmatrix} h_0 & h_1 & h_2 & \cdots & h_{n-1} \\ h_1 & h_2 & h_3 & \...
eepperly16's user avatar
  • 7,312
1 vote
0 answers
133 views

Inverting a Hankel matrix with factorial element growth

A problem in theoretical physics has led me to consider a Hankel matrix $H$ whose elements are given by $H_{ij} = (i+j-1)^{i+j-1}$, where $i, j = \left[1, N\right]$ for any $N$ you like. I'm curious ...
miggle's user avatar
  • 285
11 votes
0 answers
385 views

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 $V$ ...
eepperly16's user avatar
  • 7,312
0 votes
0 answers
81 views

Lower bound formulas of smallest eigenvalues of Hankel matrices

Are there simple and reliable formulas (using matrix elements and matrix trace) of lower bound for the smallest eigenvalue of a matrix of the following type? It's Hankel matrix
ayr's user avatar
  • 731
0 votes
0 answers
198 views

How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?

Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
JoJo's user avatar
  • 1
1 vote
0 answers
36 views

Is there an expression for Hankel minors in terms of skew Schur polynomials?

There are known expression for Toeplitz minors in terms of skew Schur polynomials, see the paper entitled ''Toeplitz minors'' by Bump and Diaconis, or e.g. 1705.08067 and 1706.02574 In particular, ...
WLV's user avatar
  • 139
1 vote
0 answers
504 views

vandermonde decomposition of a Hankel matrix

Suppose, I have a Hankel matrix H of rank-K and size L x M, I want to decompose H into vandermonde decomposition as: \begin{equation} H = SCT^T \end{equation} where S and T are vandermonde matrices of ...
Neuling's user avatar
  • 57
1 vote
1 answer
217 views

Determinant of the $n \times n$ exchange matrix

I have to find the determinant of an $n \times n$ exchange matrix. After applying values to $n$, I found out the determinant has a periodicity of $4$ and its formula is $(-1)^{n(n-1)/2}$. Literature ...
Shambhala's user avatar
  • 991
0 votes
0 answers
51 views

Eigenvalues when perturbed along anti-diagonal.

Given a $n \times n$ matrix $A$ with eigenvalues $\lambda_k$ for $k = 1, 2, \dots, n$ we know the relationship between $\{\lambda_k\}$ and the eigenvalues of $A + t I$ where $I$ is the identity matrix ...
ITA's user avatar
  • 1,833
5 votes
2 answers
230 views

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 \...
Betelgeuse's user avatar
1 vote
0 answers
90 views

Schur-Positivity of a simple polynomial

Let $\chi_{d,p;f}$ be the following symmetric polynomial, $$\chi_{d,p;f}(x)=\prod_{l=1}^d\sum_{k=0}^p x_l^{f_k},$$ where $f=\lbrace f_0,\ldots,f_p\rbrace$ is a set of integers. I need to identify for ...
Nicolas Medina Sanchez's user avatar
1 vote
1 answer
296 views

Is this Hankel matrix positive definite?

It's an exercise given in my book which says that Question: Consider a matrix $A=(a_{ij})_{5×5}$, $1\leq i,j \leq 5$ such that $a_{ij}=\frac{1}{n_i+n_j+1}$, where $n_i,n_j\in\mathbb{N}$. Then in which ...
Ibrahim Islam's user avatar
1 vote
0 answers
541 views

Determinant of a matrix where every row other than first is a cyclic shift of the first row

How do you find the determinant of the following matrix? \begin{pmatrix} 1&2&3&\cdots&n\\2&3&\cdots&n&1\\3&\cdots&n&1&2\\\vdots&\vdots&\vdots&...
GinGin3203's user avatar
3 votes
3 answers
184 views

Find the determinant of the following $5\times 5$ real matrix:

Let $A \in \mathbb{R}^{5 \times 5}$ be the matrix $$\begin{bmatrix} a&a&a&a&b\\a&a&a&b&a\\a&a&b&a&a\\a&b&a&a&a\\b&a&a&a&...
Noy's user avatar
  • 777
3 votes
2 answers
1k views

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 ...
d.y's user avatar
  • 649
1 vote
0 answers
232 views

Eigenvalues of three anti diagonal Hankel matrix

Is there a way to obtain eigenvalues of three anti-diagonal $n \times n$ Hankel matrix $$ H=\left( \begin{...
Pavel's user avatar
  • 11
3 votes
4 answers
3k views

Determinant of the identity matrix with columns in reverse order

If $C \in M_n(\mathbb{R})$ such that $(C)_{ij} = (I)_{(i)(n+1-j)}$, how do I prove that $\det (C) = -1$. What I tried: I know that $\det(C) = -1$ for $C\in M_{2}(\mathbb{R})$, but I don't know how to ...
Raton's user avatar
  • 729
3 votes
1 answer
1k views

Notation for the reverse identity matrix

I'm wondering if there's a canonical notation for the reverse identity matrix, i.e. $$ ?=\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 1\\ 0& 0 & 0 & 1 & 0\\ 0 & 0 &...
Dac0's user avatar
  • 9,254
18 votes
2 answers
1k views

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 ...
Vim's user avatar
  • 13.7k
4 votes
1 answer
770 views

Proximal operator for the nuclear matrix norm of Hankel matrix

I have a problem in hand for which I need to compute the proximal operator of the composite function $ {\left\| \mbox{Hankel} (x) \right\|}_{\ast} $ where $ x \in \mathbb R^N $ and $ \left\| \cdot \...
mmn61's user avatar
  • 41
37 votes
3 answers
2k views

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}\\...
Shevliaskovic's user avatar
2 votes
1 answer
1k views

Determinant of a Hankel matrix

Is there a closed form for determinant of the Hankel matrix of the sequence $\{ 1, a, a^2, \dots, a^{2n+2} \}$?
Sunni's user avatar
  • 4,546