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16 votes
2 answers
4k views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as follows $$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$ What is known about the singular values $\sigma_1 \geq \cdots \geq \sigma_n$ of $H$? ...
alext87's user avatar
  • 3,217
7 votes
3 answers
1k views

Checking positive semi-definiteness of integer matrix

Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements? I have to check positive semi-definiteness of many symmetric ...
SIM2's user avatar
  • 73
7 votes
0 answers
197 views

A special eigenvalue problem

For my research I need to solve a generalised eigenvalue problem $Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
yarchik's user avatar
  • 492
5 votes
2 answers
6k views

Computing the largest eigenvalue of a very large sparse matrix

I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue ...
Gadi A's user avatar
  • 233
4 votes
0 answers
2k views

What is the time complexity of the largest singular value and its vectors?

Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
B. Arsic's user avatar
  • 123
2 votes
1 answer
1k views

Is it faster to compute eigenvalues or coefficients of characteristic polynomials?

Given $A \in \mathsf{M}_n(\mathbb{C})$ (no special structure) is it (generally) faster to compute its eigenvalues or the coefficients of its characteristic polynomial? References/insights would be ...
Pietro Paparella's user avatar
2 votes
0 answers
52 views

Large-scale projected minimum-eigenvalue computations

I am interested in efficient numerical procedures for solving large-scale instances of the following projected minimum-eigenvalue problem: $$\mu := \min_{v \in \mbox{ker}(A)} \frac{v^T H v}{\lVert v \...
David Rosen's user avatar
1 vote
1 answer
331 views

Eigenvalues of a circulant: DFT or Inverse DFT Convention?

Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
ACR's user avatar
  • 879
1 vote
0 answers
179 views

QR algorithm for eigenvalues and eigenvectors of large symmetric matrices

I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
Daniel Belaish's user avatar
1 vote
0 answers
19 views

Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices

Let $\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$. $\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
Saurabh Agrawal's user avatar
0 votes
0 answers
232 views

How to analyse the range of eigenvalues of a symmetric and indefinite matrix?

Let $G$ be a symmetric and indefinite matrix defined as follows $$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$ where $S$ is a symmetric positive definite matrix of size $...
Nxy's user avatar
  • 1