Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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Convergence rate of singular values of Gaussian matrix

Say that $W$ is a $m\times 1$ vector distributed $\mathcal{N}(0,\sigma^2 I)$. Also, $X$ is a $n\times m$ Gaussian marix, $n<<m$, that is independent of $W$ with iid $\mathcal{N}(0,1)$ entries, ...
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Probability matrix [closed]

What is the probability that the rank of the random matrix of size n x n generated by py is n? Please note that the probability that this rank is 0 is 1/(n+1)^n
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Do we have the upper bound or the distribution of the following ratio?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
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Concentration inequality of the $L^2$ norm of weighted vector with moment of eigenvalues of GOE

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with ...
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1 answer
563 views

How to get the lower bound of the following $\tau$?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
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2 votes
2 answers
84 views

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent). Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...
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The ratio of spectral edge of the GOE matrix

Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda_1|>|\lambda_2|\ge \dots \ge |\lambda_n|$, can we get the order of $|\lambda_1|/|\lambda_2|$ or even $\lambda_1/\lambda_2$? Any ...
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Finding the resolvent of two coupled random matrix system using supersymmetry

I'm trying to follow the result in this work. Let me briefly introduce the problem. I have a $2N\times 2N$ matrix $$ H=\begin{pmatrix}H_1 & V \\ V^{\dagger} & H_2\end{pmatrix}. $$ Here both $...
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Is $\mathbb E [zz^T / z^Tz] = I / n$ is generally well known equation when $z \sim N(0,I)$, $I \in \mathbb{R}^{n \times n}$?

I found that following equation holds for random vector $z \sim N(0,I)$ : $\mathbb{E} [\frac{zz^T}{z^Tz}] = \frac{1}{n} I$ Proof is very simple that is only calculating integral for each component ...
2 votes
2 answers
200 views

Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
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2 votes
1 answer
66 views

Distribution of scaled Johnson-Lindenstrauss transforms

Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that $$ \...
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Do we have the universal property of the edge of the spectrum for the Wigner matrix?

In Chapter 3 of the textbook: An Introduction to Random Matrices, we have that for normalized GUE/GOE/GSE and ordering its eigenvalues $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$, we have that $$ ...
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$\mathbb{P}(\|A^Tx\| \ge \epsilon \|A\|\|x\|) \ge \delta$ for all $x \in\mathbb{R}^n$

Let $r$ and $n$ be integers such that $1 \le r \ll n$, and $\|\cdot\|$ denote the Euclidean norm of vectors or the spectral norm of matrices. Suppose that $\mathcal{D}$ is a probability distribution ...
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2 votes
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Expected value of $\mathrm{tr}((X + D)^{-1})$ where $X$ is Wishart and $D$ is diagonal?

Let $X$ be a standard Wishart matrix, i.e., $$ X = \sum_{j=1}^n g_j \otimes g_j \quad \mbox{where} \quad g_j \sim N(0, I_d). $$ Above, $g_j$ are independent samples from the standard multivariate ...
5 votes
2 answers
298 views

Is there a 'natural' projection from $O(n)$ into $S_n$?

Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties? $F(P_\sigma) = \sigma$ for all $\sigma \in S_n$ $F^{...
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1 vote
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Mixed moments of traces

I've seen a host of results concerning computations for $$\mathbb{E} \left[ \operatorname{tr} A^{i_1}\cdots \operatorname{tr} A^{i_j} \,\overline{\operatorname{tr} A^{k_1} \cdots \operatorname{tr} A^{...
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3 votes
1 answer
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Asymptotic results for smallest gap of Gaussian random matrix

For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$. ...
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1 vote
1 answer
163 views

How to prove that upper bound of the hitting time holds with high probability?

Let $G$ be a symmetric Gaussian random matrix with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=\frac{1}{n}$, and ordering its eigenvalues $\lambda_1\le \lambda_2\le \dots \le \lambda_n$ corresponding ...
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2 votes
1 answer
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The singular values of truncated Haar unitaries

I've been playing around numerically with Haar random $\text{CUE}$ unitary matrices of size $N$ by $N$, with $N$ around $1000$. If I "truncate" the matrix by keeping the upper left $fN$ by $...
2 votes
1 answer
76 views

Eigenvalues of $H_1 H_2 H_1$, where $H_1$, $H_2$ independent $\mathit{GUE}$

Given $H_1$ and $H_2$ i.i.d. $\mathit{GUE}$ matrices, what is the single eigenvalue distribution of $H_1 H_2 H_1$ in the large $N$ limit? This matrix is Hermitian, and so its eigenvalues are still ...
2 votes
1 answer
158 views

Resolvent (Green's function) of this random matrix

I have a matrix $A$ as follows: $$ A=\begin{pmatrix} 0 & \boldsymbol{W} \\ \boldsymbol{W}^{\dagger} & \boldsymbol{H} \end{pmatrix} $$ where $H$ and $W$ are a random Hermitian $N\times N$ ...
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1 answer
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What is the non-asymptotic upper bound for the leading eigenvector of the random matrix?

Fix a Gaussian random matrix $A$ with $E[A_{ij}]=0$ for $i, j=1,\dots n$ and $E[A_{ij}^2]=\frac{1}{n}$. Let $v_1$ be the leading eigenvector of $A$. What is the non-asymptotic upper bound for $v_1$, ...
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3 votes
1 answer
63 views

Interpolating Wigner's semicircle and Girko's circular law

I am relatively new to the field of random matrices, and I suspect this may be relatively well-known. Consider the real $N$ x $N$ matrix $O$ with i.i.d. standard normal entries, and consider the ...
1 vote
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58 views

Integration with respect to Haar measures normalised over a subspace

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work $\int_{\mathcal{U}(d)} \frac{\...
2 votes
1 answer
65 views

What condition on random matrix can preserve sub-Gaussian property?

Suppose $x\in SG(\sigma^2)$ is a sub-Gaussian random vector, i.e. $\left<u,x\right>\quad \forall u\in \mathbb{S}^{n-1}$ is a sub-Gaussian random variable. My question is : what condition on the ...
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10 views

Hoeffding type bound for functions of eigenvalues of Wigner matrix

Let $X$ be a random $d \times d$ Wigner matrix with entries of variance $1/d$ (that is, entries are i.i.d; for simplicity we can consider a GUE matrix). Is there a concentration bound on functions $f(\...
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36 views

Integral of elements of random unitaries

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function: $$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
1 vote
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73 views

Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries

Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties: $M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative). The diagonal entries of $M$ ...
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29 views

Limiting value of normalized trace of $A^{-1} e^{-tW} A e^{-tW}$, where $W$ is Wishart matrix and $A$ is deterministic

Let $n$ and $d$ be large positive integers and let $\gamma \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid entries from $N(0,1/d)$. Let $A$ be an invertible deterministic $d \times ...
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3 votes
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Expected norms of Wishart matrices

Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below? $$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^...
2 votes
1 answer
65 views

Induced distribution from random unitary

I have a vector space which is a tensor product of two vector spaces, of dimensions $d_1, d_2$ respectively. Consider Haar random unitaries acting on the full space with matrix elements $U_{i_1 j_1, ...
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8 votes
1 answer
287 views

Why impossible events have some drawbacks or pathologies in probability theory?

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
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Bound the conditional expectation of a random matrix under weak dependence

Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\...
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Free probalities and random matrices : on the asymptotic joint behavior of non self-adjoint random matrices with non Gaussians entries

One of the famous theorem in random matrix theory is one termed circular law. It is stated that if $a_{ij}$, for all $1\leq i\leq n, 1\leq j\leq n$,, is a famille of $i.i.d.$ complexe centered random ...
1 vote
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123 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 $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$ ...
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Characterization of the extreme eigenvalue of Wishart-Laguerre/Jacobi-MANOVA ensemble

The Tracy-Widom distribution gives the limiting distribution of the rescaled largest eigenvalue of a random matrix taken from an appropriate symmetry class. According to Bloemendal, the deformed Tracy-...
3 votes
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124 views

Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
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79 views

Correlation between vectors after matrix-multiplication

Consider $x$ and $y$ two $N\times 1$ complex vectors, and $T$ a $N\times N$ complex random matrix. Each element of $T$ is chosen in a complex normal distribution. Let us also define the (Pearson) ...
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1 answer
148 views

Spacings of Satake parameters under Ramanujan conjecture

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match: the distribution of spacings between Satake parameters of an L-function $F$ ...
1 vote
1 answer
158 views

Using gradient descent in probability case

Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of: $$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...
8 votes
0 answers
207 views

Decay of orthogonal contributions in a random set of vectors

Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$: $$\frac{v_1}{\|v_1\|},\...
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28 views

Computing the expectation of $\mathrm{tr}((W + A)^{-1})$ where $W$ is a random Wishart matrix?

Let $W \sim \mathcal{W}_d(V, n)$ be a random Wishart matrix. Let $A$ be a real symmetric positive definite matrix. I am interested in computing $$ \varphi_{V, d, n}(A) := \mathrm{tr}\big[\mathbb{E}[(W ...
1 vote
2 answers
97 views

How to compute the trace of this random matrix: $\mathrm{Tr} ( a Y_1+ b Y_2)^{-1} ( c Y_1+ d Y_2)$?

Lets say $Y=\frac{1}{n}XX^\intercal$ and $X$ is a $n\times m$ random matrix whose entries are i.i.d gaussian. We know when $n$ and $m$ go to infinity with a fixed ratio, the singular values of $Y$ ...
1 vote
1 answer
63 views

Singular values of a Gaussian random times deterministic diagonal matrix

Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...
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3 votes
1 answer
300 views

Is this combinatorial identity known? (of interest for random matrix theory)

While playing around with random matrices and I arrived at a different formula for the mean of the limiting normal distribution for a spectral CLT for sample covariance matrices. More precisely I have ...
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3 votes
1 answer
110 views

What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?

The Tracy–Widom distributions admit many interpretations. One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...
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0 votes
1 answer
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On the invertibility of $Z^\top Z$, where $Z$ is a Random matrix with concentrated weakly correlated entries

Let $d$, $n$, and $m$ be large positive integers. Let $X=(x_1,\ldots,x_n) \in \mathbb R^{n \times d}$ be a random matrix iid rows from some distribition $P$ on $\mathbb R^d$ which admits a density. ...
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1 vote
1 answer
76 views

Expectation value of random GUE matrix

Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors. I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value $$\...
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2 votes
1 answer
199 views

Upper-bound for spectral norm of the covariance matrix of a certain Gaussian vector with correlated entries

Let $n$ and $m$ be large positive integers. Let $x=(x_1,\ldots,x_n)$ be a vector of independent random variables from $N(0,1)$. It is clear that the covariance matrix of $x$ is $I_n$, the identity ...
  • 6,094
1 vote
0 answers
43 views

Characterizing set of IID average of symmetric positive semidefinite matrices matrices

Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices. Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...

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