All Questions
34 questions
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 $H$?
...
- 3,217
12
votes
4
answers
904
views
Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts
Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix
...
- 197
9
votes
1
answer
396
views
Bound on the ratio of top 2 eigenvalues
Let $P$ be a $n \times n$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = 1 - (n-1)\tau$ where $0<\tau < \frac{1}{n}$.
It is clear that the largest eigenvalue of $P$ is 1, ...
user31317
7
votes
1
answer
2k
views
Bound the eigenvalue of product of matrices?
Let $H$ be a $n \times n$ real symmetric matrix that has eigenvalues with absolute value less than 1. Define the matrix $M = \prod_{i=1}^n (I - e_ie_i^{\top}H)$ where $e_i$ denotes the $i^\text{th}$ ...
user31317
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222
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{...
- 492
6
votes
1
answer
879
views
A question on the smallest singular value
Let $X(r)$ be the set of matrices $A \in M(n \times m)$, $n \leq m$, such that the norm of $A$ (largest singular value) is smaller or equal than $1$ and the smallest singular value of $A$ is smaller ...
- 61
6
votes
0
answers
10k
views
Eigenvalues of the sum of two matrices [duplicate]
I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B = C$ as a function of the eigenvalues of $A$ and of the eigenvalues of $B$ is generally a non-easy problem. I was ...
- 61
5
votes
1
answer
375
views
Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?
I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...
- 1
5
votes
1
answer
417
views
Log determinant of quadratic form
I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
- 53
4
votes
2
answers
477
views
Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
- 181
4
votes
1
answer
206
views
How to find the analytical representation of eigenvalues of the matrix $G$?
I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
- 153
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
4
votes
0
answers
284
views
Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
- 2,712
4
votes
1
answer
4k
views
Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$
I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...
- 297
3
votes
1
answer
4k
views
Eigenvalues of the sum of a positive matrix and a positive rank-$1$ matrix
Assume that $A, B$ are positive $n \times n$ matrices and that $B$ is rank-$1$, i.e., $B=xx^*$. If the eigenvalues of $A$ are $a_1 \geq a_2 \geq \cdots \geq a_n$, and $x$ is not an eigenvector of $A$, ...
- 31
3
votes
1
answer
264
views
When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
- 2,239
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
2
votes
1
answer
74
views
Limitation through the singular values
Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
- 143
2
votes
1
answer
345
views
Alike looking matrices imply convergence of eigenvalues?
This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...
user54300
2
votes
1
answer
359
views
Dimension independent computational complexity of singular value decomposition
Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...
- 45
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 145
1
vote
1
answer
506
views
Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues
For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset
$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$
is a manifold of dimension $2n(2r)-(...
- 143
1
vote
1
answer
331
views
Eigenvalues of a circulant: DFT or Inverse DFT Convention?
Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
- 879
1
vote
1
answer
467
views
Trace minimization for generalized eigenvalue problem
In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min_{Y \in Y^*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum_{i=1}^p \lambda_i,
$$
with
$$
\text{
$X^...
- 21
1
vote
1
answer
223
views
Growth rates of ‘sub-traces’ of matrices
Consider an aperiodic, non-negative and non-zero $m\times m$ matrix $A$. Here aperiodic means that there exists an integer $n$ such that every entry of $A^n$ is strictly positive. By the Perron–...
- 89
1
vote
0
answers
163
views
An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$
Let us consider the following matrix $A=(a_{k,l})$ where
$$A=\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$$
Let us consider the submatrix $A_0$ of $A$ whose entries are those $a_{k,l}$ where $k\...
- 4,058
1
vote
0
answers
356
views
Is there relation between eigenvalues of A,B,A+B,AB? [closed]
I found there are some works about eigenvalues of $A,B,A+B$ in the case of Hermitian matrices. For example, here.
I am wondering in the case that $A$ and $B$ is not Hermitian.
For example, my question ...
- 405
1
vote
0
answers
132
views
Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
1
vote
0
answers
34
views
A recap of regularity of singular values as a function over M_n
So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$
$$A \rightarrow s_n(A) $$
where $s_n(A)$ is the greatest singular value of A. I know there has ...
- 245
1
vote
0
answers
148
views
Perturbation of eigenvalues of some special matrices
In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
- 61
0
votes
0
answers
224
views
Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle
Consider the matrix
$$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$
where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
- 101
-1
votes
1
answer
230
views
How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]
(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...
- 181