# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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### 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 ...

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43 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 ...

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66 views

### Expectation of the inverse of random principal submatrices

The goal of this question is finding the concentration point of the inverse of random principal submatrices, which is posed as follows. Consider $\mathbf{S}\in\mathbb{S}^{n}_{++}$ to be a strictly ...

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43 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 ...

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49 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 ...

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96 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 ...

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133 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 ...

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**2**answers

119 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\...

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20 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 ...

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89 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 ...

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136 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 ...

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78 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 ...

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59 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}
$...

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71 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\...

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59 views

### Singular value of hadamard product

Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0, Var(A_{i j}) = 1/\sqrt n$ for any $i,j$. B is an $n \times n$ symmetric matrix with $B_{ii} = 0$.
I need to find a upper bound ...

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88 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 ...

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49 views

### Independence of random projection and orthogonal projection

Suppose we have three fixed unit vectors $x, y, z \in \mathbb{R}^d$ and an (arbitrary) distribution over random matrices $M \in \mathbb{R}^{k \times d}$: let $P_M = M^T(MM^T)^{-1}M$ and $P^{\perp}_M = ...

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89 views

### Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices

Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...

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88 views

### Bound on $i$th largest eigenvalue in a large Erdos-Renyi graphs

Typical magnitude of $i$th largest eigenvalue of an Erdos-Renyi random graph seems to decay at least exponentially with $i$. Is there an analytic expression for the constant in the exponent, or a nice ...

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184 views

### Subspaces with all vectors having large $\|x\|_{\infty}/\|x\|_2$ value

I am able to show that any $k$-dimensional subspace of $\mathbf{R}^{Ck\log(k)}$ must contain a unit vector $x$ such that $\|x\|_{\infty} \ge c\sqrt{1/\log(k)}$ for a small enough constant $c$.
But is ...

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175 views

### Factorisation of Gaussian random matrix into random Hermitian and correction factor

By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...

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171 views

### Moments of inner products for Haar random matrices

Let $\mathbf{U}$ be a $n \times n$ Haar orthogonal matrix, $\mathbf{D}$ be a fixed diagonal matrix with half of its entries $+1$ and the remaining half $-1$ and $\left \langle\cdot, \cdot \right \...

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107 views

### Random sequence with positive Lyapunov exponent?

Consider the following self-adjoint matrix
$A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some ...

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70 views

### What is the distribution of eigenvalues of $A^TA$, where $A \sim N(\mu, \Sigma)$?

Let $A$ be a random matrix following multivariate normal distribution $N(\mu, \Sigma)$.
What is the distribution of the eigenvalues of $A^TA$?
A reference to the literature would be most welcome.

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69 views

### Dimension-free sample complexity for estimating Gaussian covariance

(also asked on math.se, with no answers)
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$$...

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125 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 ...

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69 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{...

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81 views

### Maximum volume submatrices of a Khatri-Rao product of matrix exponentials

My question requires quite a bit of setup, which leads to a conjecture. So I split my question into three parts, Setup, Conjecture, and Question.
Setup:
Pick any two right stochastic matrices $\...

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323 views

### Use statistical physics ideas ("replica trick") to compute asymptotic value of $\inf_{\|w\| \le r} (1/n)\|Xw-y\|^2$ for random $X$ and $y$

I'm trying to get my head around the "replica trick" and it's mathematically rigorous formulations (due to Talagrand, Parchenko, etc.). I was wondering to myself that a solution or insight ...

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74 views

### Difference between identity and a random projection

Suppose a random projection $P$ in $\mathbb{R}^d$ onto a random n-dimensional subspace in $\mathbb{R}^d$ uniformly distributed in the Grassmannian $G_{d, n}$ (the projection of the row space of a ...

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174 views

### Dyadic distribution of $0/1$ permanents

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.
What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents
in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...

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14 views

### Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows.
We wish to find a tight bound on the singular values. We know on the rows that they have (...

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31 views

### condition number of random submatrices

If we randomly pick $k\ll n$ columns from a fixed $n\times n$ matrix $A$, what can one say about the distribution of the 2-norm condition number of the resulting $n\times k$ matrices $A_k$?
I'd expect ...

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**1**answer

77 views

### Eigenvalues of large symmetric random tensors

I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting ...

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22 views

### Covariance concentration bound for randomly sampled positive semi-definite matrices

I saw the following inequality being used in a paper and the given reference was Joel A Tropp et al. An introduction to matrix concentration inequalities. However, I could not find this inequality ...

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276 views

### Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...

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106 views

### Concentration inequality for the sample covariance matrix

I'd like to know if there is a concentration inequality for the sample covariance matrix that don't assume the knowledge of the true mean.
Background.
Given a probability distribution $\mu$ on $\...

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146 views

### Spectral gap of $AA^{T}$ for Bernoulli random matrix A

I need the following answer for research purposes.
Let $A$ be a $m \times n$ random matrix with iid ${\rm Bernoulli}(p)$ entries. Is there any result on the spectral gap of $AA^{T}$ (similar to well ...

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34 views

### Spectral properties of random principal submatrices

Let $M$ be a matrix. A principal submatrix of $M$ is a square matrix obtained by deleting from $M$ the $i$-th column and the $i$-th row, for some number of indices $i$ (the rows and columns are ...

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67 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, ...

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45 views

### Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix

Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...

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63 views

### Lower bound of the probability of singular random matrix over $\{\pm1\}$ in ``Singularity of random Bernoulli matrices"

Suppose $M_{n}$ is an $n \times n$ matrix with independent ±1 entries. Recent breakthrough shows that the probability $\mathbb{P}(M_{n} \text{ is singular})$ is
$$(1) \quad\quad\qquad \mathbb{P}(M_{n} ...

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98 views

### Combinatorial formula to compute the moments of the product of two free random variables

I found in the PhD thesis Moments method for random matrices with applications to wireless communication the following combinatorial formula to compute the free moments of the product of two random ...

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67 views

### Eigenvalues of Hadamard product of two Wishart-type matrices

Given two independent Gaussian matrices with i.i.d. entries: $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times q}$, where and $A_{i,j},B_{i,j}\sim\mathcal{N}(0,1)$. Assume that $\max(p,q)<n....

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36 views

### Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...

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132 views

### How to compute the first moment of the distribution of the convolution of Marcenko-Pastur law with a not iid matrix?

Let $\mathbf{F}$ denote an M × N matrix whose entries are independent zero-mean complex random variables, the limiting eigenvalue distribution is given by the Marchenko Pastur law $MP_{\beta}$, where $...

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93 views

### Distribution of eigenvectors of random matrices and link with the components of the matrix

Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0.
Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\...

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**1**answer

50 views

### General formula for the integral w.r.t to Marchenko-Pastur density, of the ratio of degree $\le 2$ polynomials

Question. Is there a closed-form formula (via standard objects like rational functions, radicals, special functions, special numbers like Catalan numbers, etc.) expressing integrals of rational ...

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157 views

### Expectation of product of random matrices

Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?
In particular, ...

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99 views

### Limiting eigenvalue distribution of $YY^\top$ where $Y_{ij} = X_{ij} + a$ and $X$ has iid rows from an isotropic log-concave distribution

Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$ such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ ...