All Questions
19 questions
3
votes
2
answers
2k
views
Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
- 2,252
8
votes
3
answers
509
views
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
- 484
1
vote
0
answers
148
views
Eigenvalue distribution of random matrices
Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider
$$\sum_i x_i M_i$$
for random variables $x_i$.
What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$
...
- 63.9k
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 $\...
- 7,514
2
votes
1
answer
236
views
How can I prove a randomly generated matrix has distinct non-zero eigenvalues?
Consider the following $M×M$ matrix
$$
\mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K)
$$
where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
- 52.3k
2
votes
1
answer
342
views
Two-level correlation function of eigenvalues for large random matrices
One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as:
\begin{equation}
\rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle
\end{...
- 188k
1
vote
1
answer
174
views
Probability finite precision random matrix has distinct eigenvalues
copied from math stack exchange
There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, ...
- 63.9k
2
votes
1
answer
668
views
Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...
- 6,853
1
vote
1
answer
391
views
Calculation of the variance in the Wishart random matrix ensemble
I'm trying to obtain the mean and variance of a function given by $f=\sum_{i=1}^{N} a \lambda_i$, where $a$ is a constant and $\lambda_i$ are the ordered eigenvalues of a Wishart matrix given by $\...
- 188k
0
votes
1
answer
150
views
Expectation of random matrix
Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that
\begin{align}
E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
- 128k
2
votes
1
answer
265
views
Limit law of eigenvalue of random matrix with mean different to 0
If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let
$$Y = X X^T$$
with $X^T$ the ...
- 63.9k
5
votes
1
answer
3k
views
Eigenvalues and eigenvectors of Gaussian random matrices
Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues?
...
- 5,429
3
votes
0
answers
184
views
Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion
I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
- 41
11
votes
8
answers
2k
views
Semicircle law universality elsewhere
Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
- 188k
4
votes
1
answer
372
views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
- 188k
4
votes
2
answers
832
views
Spectra of VERY sparse random matrices
Consider an $n\times n$ random binary matrix $M$ with i.i.d. entries $m_{ij} \sim {\rm Bernoulli}(p)$, where $p = n^{-\beta}$ with $\beta \in (1,2)$. I am interested in the behavior of the singular ...
1
vote
0
answers
146
views
minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
- 363
5
votes
0
answers
327
views
Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
- 161
2
votes
0
answers
756
views
Distributions of eigenvalues for matrix normal distribution: related references
I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution.
I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...
- 169