Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
841
questions
5
votes
2
answers
773
views
Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$,
and $np\to c=2,3$.
The plots above are already ...
2
votes
0
answers
180
views
Harish-Chandra–Itzykson–Zuber integral with two terms
We know
$$
\int \mathcal{D}U \exp(\mathrm{Tr}(AUBU^*))
$$
can be computed by Harish-Chandra–Itzykson–Zuber(HCIZ) integral. I am wondering whether it is possible to compute
$$
I=\int \mathcal{D}U \exp(\...
3
votes
0
answers
259
views
Gaussian integral with Vandermonde determinant
I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant:
$$
\int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x),
$$
...
6
votes
0
answers
206
views
Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$
We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
0
votes
1
answer
162
views
LDP for Marchenko Pastur with k/n tending to 0
I am interested in the determinant of $W = X * X'$, where $X \in \mathbb{R}^{k \times n}$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $\mathbb{R}^{n}$. (I am aware of ...
1
vote
0
answers
107
views
Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
3
votes
0
answers
91
views
Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)
For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
5
votes
0
answers
155
views
Determinant bounded away from zero
Suppose $P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1})$ is a noncommutative polynomial with positive coefficients. We may then consider the map $g:U(n)^k\rightarrow\mathbb{C}$ from the unitary group $U(n)$,...
1
vote
1
answer
146
views
Estimates of product of eigenvalues gaps for Wigner matrices
Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$.
My question. For any fixed $i\in\{1,\dots,n\}$, ...
5
votes
1
answer
503
views
Riemann-Hilbert approach to Selberg integral
I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
0
votes
1
answer
155
views
Bernoulli random matrix and rotationally-invariant ensemble [closed]
I am recently reading about random matrix theory in engineering applications. I come up with the following question. I have been trying to find any references but it doesn't help. Hopefully, anyone ...
0
votes
0
answers
84
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
7
votes
4
answers
454
views
What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?
Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$...
2
votes
1
answer
122
views
Are the eigenvalues of the 1D lattice with random weights known?
Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge ...
6
votes
0
answers
327
views
Typical eigenspectrum of a random projection of a large matrix
Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection
$$ A =...
2
votes
0
answers
109
views
What is this particular distribution in random matrix theory?
In a certain statistical problem involving large random matrices, I recently encountered the distribution on the picture below. I did not manage to identify it after browsing the internet for quite ...
1
vote
1
answer
179
views
Orthogonal transformation of multivariate Bernoulli-Gaussian distribution
Actually, I have asked this question in https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution, but I think mathoverflow might be ...
1
vote
1
answer
97
views
Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$
Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
$\mbox{trace}(\Sigma_d) = 1$.
$\|\Sigma_d\|_{op} = \mathcal ...
1
vote
1
answer
120
views
Moments of rescaled Bernoulli random matrix
Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$
Is ...
1
vote
0
answers
154
views
Largest nuclear norm of $n \times n$ symmetric matrices whose entries are between -1 and 1
Let $\cal M$ be the set of real symmetric $n \times n$ matrices whose entries are all in the interval $[-1, 1]$. I'm interested in understanding the largest possible nuclear norm of these matrices as ...
1
vote
0
answers
77
views
Random matrix heuristics for Koopman operators
Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator
$$\mathcal{K} : \ f \mapsto f \circ T$$
has a nice spectrum ...
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
1
vote
1
answer
120
views
Given iid $w_1,\ldots,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_{op} \to 0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\ldots,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
0
votes
0
answers
121
views
Spectral CLT for random matrices with iid entries
Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\...
6
votes
1
answer
727
views
Expected value of orthogonal projection $X^{+}X$
Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\...
1
vote
0
answers
97
views
Exponential decay of a random matrix falling into a ball
Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
3
votes
1
answer
519
views
Top singular value of large random matrices: concentration results
Let $A$ be a $n\times m$ random matrix, whose elements $a_{ij}$ are independent standard Gaussian random variables.
I am interested in the case $n=\alpha N\,$, $\,m=(1-\alpha)N$ for $\alpha\in(0,1)$ ...
4
votes
1
answer
202
views
Limiting eigenvalue distribution of $(I-A)^T(I-A)$
Let $A\in\mathbb R^{n\times n}$ be a random Gaussian matrix with i.i.d entries from $\mathcal N (0, \frac{a}{\sqrt{n}})$. By Marchenko-Pastur we know the limiting distribution of the eigenvalue of $A^...
2
votes
0
answers
116
views
Limiting PDF of the eigenvalue of random Gaussian matrix
It has been proven that the CDF of the eigenvalue distribution of random Gaussian matrix converges to a uniform disk circular law. Is it true for the PDF of the limiting eigenvalue distribution? In ...
4
votes
1
answer
201
views
Expected even power of absolute value of an element of a random unitary matrix
Let $t$ be a natural number. For a unitary matrix $U$ let $U_{1,1}$ be the top left matrix element of $U$. I am trying to figure out the value of $\int |U_{1,1}|^{2t} dH(U)$ where $H$ is the Haar ...
4
votes
2
answers
225
views
A 'projective' property of the Haar U(n) measure
Let $U(n)$ be the compact manifold of unitary $(n \times n)$-matrices and let $\mu_n$ denote the Haar-probability measure on $U(n)$. For $m < n$ does there exists a measurable (maybe even ...
6
votes
1
answer
248
views
L^1-norm of the trace on the unitary group
Is the value of the following integral over the unitary group with respect to the normalized Haar measure known?
$$
\int |Tr(U)|^pdU.
$$
There are some results for $p=2k$, $k\in\mathbb N$ (see ...
1
vote
1
answer
367
views
Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
2
votes
1
answer
85
views
Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix
Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
5
votes
0
answers
230
views
Expected value of $X^{\top}(XAX^{\top})^{-1}X$ for large random $X$
Let $X\in \mathbb{R}^{m\times n}$ be a random matrix where the entries are i.i.d. standard normal, and let $A\in \mathbb{R}^{n\times n}$ be a deterministic diagonal matrix with positive entries on the ...
1
vote
0
answers
79
views
How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]
Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$.
I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$.
My belief is that this is true is motivated by empirical ...
1
vote
0
answers
213
views
Lower bound for expectation of minimum eigenvalue
Let $X$ be a random (symmetric) matrix drawn from an unknown distribution. I have an estimate of $\lambda_{\min}(\mathbf{E}[X])$. Specifically, I have
$$\lambda_{\min}(\mathbf{E}[X]) \geq c$$ a ...
1
vote
1
answer
173
views
Johnson-Lindenstrauss with Orthogonalization
I have been looking at constructions satisfying the Johnson-Lindenstrauss Lemma (e.g., projections onto random subspaces, random Gaussian matrices, random Rademacher matrices, etc.). It seems that ...
2
votes
1
answer
160
views
Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
4
votes
1
answer
302
views
Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
4
votes
1
answer
301
views
Haar integral of rational function of unitaries
I'm trying to compute the following Haar integral over the unitary group:
$$
\int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU.
$$ Is there anything known about the ...
0
votes
2
answers
363
views
Free multiplicative convolution of two random matrices
Given two freely independent random hermitian matrices $A$ and $B$ following laws $\mu, \nu$, one can compute the empirical spectral distribution of $AB$ by their free multiplicative convolution $\mu\...
2
votes
0
answers
59
views
What is the distribution of determinant of multi multiplication of some Gaussian matrices?
I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
2
votes
0
answers
103
views
The distribution of eigenvalues of linear combinations of random unitary matrices
Suppose that $\alpha_{1},\dots,\alpha_{r}$ are non-zero complex numbers. Let $U_{1},\dots,U_{r}$ be random $n\times n$-unitary matrices. Let $A=\alpha_{1}U_{1}+\dots+\alpha_{r}U_{r}$.
I have observed ...
6
votes
1
answer
323
views
Do random asymmetric games have more complicated strategies than random symmetric games?
Let $\Delta \subset \mathbb R^n$ be the locus of vectors whose entries are nonnegative and sum to $1$.
For $M$ an $n\times n$ matrix over $\mathbb R$, let $x_M \in \Delta$ be the vector $x$ that ...
1
vote
1
answer
480
views
Interpretation of Bai-Yin theorem and a question about (Hastie, Montanari, Rosset & Tibshirani)
Let $X_n\in \mathbb{R}^{p\times n}$ be a random matrix whose entries are i.i.d. $\mathcal{N}(0,1)$. Define $S_n = \frac{1}{n}X_n X_n^\top$. If $p/n\to y\in (0,1)$, the well-known Bai-Yin theorem ...
4
votes
0
answers
74
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
1
vote
1
answer
196
views
How to normalize an Inverse Wishart random matrix?
Background:
Let $d\in \mathbb{N}$.
Define the space of (real symmetric) positive definite matrices of size $d\times d$ as follows:
\begin{align}
\mathcal{S}_{++}^d := \big\{\mathbb{M}\in \mathbb{R}^{d\...
2
votes
1
answer
315
views
Singular value of Hadamard product
Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$.
I need to find a upper bound of ...
2
votes
1
answer
1k
views
Concentration of the norm of subGaussian random vectors
I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.
I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...