All Questions
Tagged with random-matrices sp.spectral-theory
27 questions
17
votes
4
answers
2k
views
An experiment on random matrices
A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...
- 9,007
16
votes
3
answers
2k
views
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 73
14
votes
1
answer
449
views
References for reasoning about the spectrum of a convex body?
By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
- 143
6
votes
1
answer
299
views
Phase transition in matrix
Playing around with Matlab I noticed something very peculiar:
Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by
$$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$
...
- 536
4
votes
1
answer
237
views
Spectral density of symmetrized Haar matrix
Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
- 1,608
4
votes
1
answer
1k
views
Expected value of the spectral norm of a Wishart matrix?
Let $x_1,\dots,x_n$ be i.i.d. drawn from $N(0,I_{p\times p})$. Consider the sample covariance matrix $W(n,p)=\frac 1n \sum_{i=1}^n x_ix_i^T$, a Wishart matrix.
For fixed $n,p$, what is the expected ...
- 947
4
votes
1
answer
359
views
Horn's spectrum problem with random Hermitian matrices
An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the
saturation conjecture. J. Amer....
- 52.3k
4
votes
1
answer
414
views
Distribution of eigenvectors and eigenvalues for random, symmetric matrix
Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.
Let $f({\bf ...
- 43
4
votes
0
answers
134
views
What is known about the density of states for the Anderson Model?
The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(\mathbb{Z}^d)$)
$$
H_\omega = - \triangle + V(\omega)
$$
where $V(\omega) \mid x \rangle = \omega(x) \mid x \rangle$ ...
- 1,054
3
votes
2
answers
307
views
Random matrix is positive
This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the ...
- 536
3
votes
1
answer
159
views
About concentration of eigenvalues values of a random symmetric matrix in a specific interval
Given a random symmetric matrix $M$ and two numbers $\lambda_\min$ and $\lambda_\max$ how do we calculate the expected or high probability value of the fraction of its eigenvalues which lie in the ...
- 2,246
3
votes
1
answer
165
views
Proving anti-concentration for the operator norm of a random matrix
If $X$ is a random matrix then I would like to find $\theta >0$ and $\delta \in (0,1)$ s.t I can say,
$$\mathbb{P} \Bigg [ \Big \vert \Vert X \Vert - \mathbb{E} [ \Vert X \Vert ] \Big \vert > \...
- 2,246
2
votes
2
answers
264
views
Is tridiagonal reduction the current best practice to compute eigenvalues of random matrices from the Gaussian ensembles (GOE, GUE, GSE)?
I have tried to compute the eigenvalues of random matrices of the GOE ensemble, using MATLAB.
Such matrices of size $n * n $ can be obtained easily, symmetrizing matrices whose elements follow the ...
- 153
2
votes
1
answer
2k
views
Bounds on the eigenvalues of the covariance matrix of a sub-Gaussian vector
Suppose that $\boldsymbol{x}\in\mathbb{R}^n$ is subgaussian random vector of variance proxy $\sigma^2$, i.e.,
$$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\...
- 127
2
votes
3
answers
999
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
2
votes
1
answer
280
views
The effect of random projections on matrices
Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...
- 1,495
2
votes
2
answers
144
views
Spectrum of finite-band random matrices?
Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...
- 245
2
votes
1
answer
276
views
Spectral norm tail bound of a correlated random matrix
I am looking for the tail bound of spectral norm for certain type of random matrix.
Let's say we have a $n\times n$ symmetric random matrix $R$, and for each entry $R_{ij}$, we have that
$$
E[R_{ij}]...
- 133
2
votes
0
answers
41
views
Why has the random Koopman matrix $ G_{xx}^{(-)} G_{yx} $ only eigenvalues on the complex unit circle?
Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution,
e.g.
$$ U_{i j} \sim N(0,1) \quad \quad i,j=1,...n+1$$
Let $ G=U U^* $ be a Gram matrix where $ U^* ...
- 253
2
votes
0
answers
116
views
Smallest singular value distribution
Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...
- 83
2
votes
0
answers
452
views
Largest eigenvalues distribution of tridiagonal symmetric random matrix
I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way.
All the ${\lambda}_i$ are distributed the same way with chi-square (...
- 21
1
vote
1
answer
455
views
Spectral theory of real symmetric matrices with random diagonal elements
Can you point me in the direction of any research done on the spectral theory (i.e. eigenvalues and eigenvectors) of real symmetric matrices with random (Gaussian or Levy) diagonal elements and fixed ...
- 221
1
vote
2
answers
892
views
Concentration of matrix norms under random projection.
Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
Are there cases where one can been able to quantify $P_G ...
- 617
1
vote
1
answer
206
views
Moment generating function of spectral norm of iid N(0,1) data matrix
Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is,...
- 113
0
votes
1
answer
622
views
Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
- 617
0
votes
0
answers
48
views
Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
- 1,755
0
votes
0
answers
160
views
$l_{\infty}$ norms of matrix perturbations
Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(...
- 1,893