# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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### 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 ...
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### If a sequence $X_n$ of RVs converges in probability to $X$, does the sequence $\mathbb{E}(X_n)$ also converge to $\mathbb{E}(X)$? [closed]

I couldn't find the answer in literature so any idea would be helpful.
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### Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$

Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
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### Upper-bound for eigenvalues of $E [UU^T]$, where $U$ is uniformly distributed on the unit $n$-sphere

Let $X$ be a $\sigma$-subGaussian random vector on $\mathbb R^n$ (for large $n \ge 3$), meaning that the random variable $X^Tv$ is $\sigma$-subGaussian for every unit vector $v \in \mathbb R^n$. ...
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### Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$

Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following: ...
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### Concentration of $2$-norms of random variables whose co-ordinates are not independent?

Let us consider the random vector $X=[X_1 \dots X_d] \in \mathbb{R}^d, E[X]= 0, cov[X]= \Sigma.$ Then the random vector $Z:= \Sigma^{-1/2} X=[Z_1 \dots Z_d]$ has $E[Z]=0, cov[Z]=I_d.$ I'm looking for ...
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### Comparison of concentrations of different $L^p$-norms of (sub) Gaussian distributions

It's well-known that the Euclidean $2$-norm of subgaussian random vectors concentrates in high dimensions, e.g. when $X \sim \mathcal{N}(0,I_n),$ (or in general $X$ is subgaussian with independent co-...
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### Singular values of random uniform matrix

Suppose $X \in \mathbb{R}^{N \times M}$ with elements sampled i.i.d. from $\mathcal{U}(-\sigma, \sigma)$. I would like to find the marginal distribution of the unordered singular values of $X$. The ...
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### Terminology: “sufficiently large absolute constant”

I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) ...
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### Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix

Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
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### Random sparse and invertible matrices

Let $n\leq m$ and $0\leq k\leq (n\times m - \min\{n,m\})$ be in $\mathbb{N}$. Let $\mu$ be a probability measure dominated by the Lebesgue measure on $\mathbb{R}$ and generate a random $n\times m$ ...
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### 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? ...
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### Reference request for concentration on measure, following Vershynin's “High dimensional Probability” book, referred often in this question

I'm new to "concentration of measure" phenomenon that I need to learn quickly (started already, but would like to pick up the remaining basic results all within a week or two to get a working ...
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### Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal

Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
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### Symplectic geometry connects random density matrices?

This question arises from studying the following papers: Christandl et al. '14 and Mejia et al. '16. These two papers use a connection between symplectic geometry and reduced density matrix. In ...
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### Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the “features” are correlated: references

Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
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### Is there any relation between moments of random matrix and its eigenvalue distribution?

Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation ...
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### Eigenvalue distribution of a random matrix

Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables ...
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### Spectrum of large random asymmetric matrices with correlation

Background: In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral: \begin{equation} I=\left[\...
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### Jacobian of changing of variables to singular value decomposition

It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues. Now suppose I have a ...
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### Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
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### Eigenvalue distribution of a band matrix

Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$. For some positive integer $k$, I define ...
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### Tighter upper bound for $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma]$

Following this question I was thinking about ways to improve the upper bound and came up with the following argument. We want to find an upper bound for \begin{equation} \mathbb{E} [\max_{\sigma \in ...
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### Minimizing the largest eigenvalue of random matrices

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition. Consider the following minimization problem:...
Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball, but outside the ball of radius $r$ for some $r \in (0,1)$, i.e. $r \leq \|X_i\| \leq 1$ ....
### Variant of Marcenko-Pastur law when the random sample always lie on a low, fixed dimensional subspace as $n,p\to \infty$?
Let our data set (or rather, sequence of datasets) be denoted by $X:=[x_1,...x_n] \in \mathbb{R}^{p\times n}$, where each datum $x_i \in \mathbb{R}^{p\times 1}$. Assume the hypothesis of Marcenko-...