Questions tagged [random-matrices]

For questions concerning random matrices.

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Generate random symmetric matrix with largest eigenvalue approximately 1

My goal is to generate a positive (entry-wise) matrix $P\in \mathbb{R}_{>0}^{N\times m}$ and then to set $S=PP^T$ such that the largest eigenvalue of $S$ is $\approx 1$ (or equal). Note that if $y$ ...
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Question in information plus noise models in random matrix theory

Hey I was hoping someone could tell me more about this topic, maybe give me some references which can help answer this question: Let $W=R+S$ be a $N \times M$ matrix with $R$ a random matrix with ...
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Multiplication of two random matrices over a finite field

Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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Tail bounds for Tracy Widom distribution

Let A be an $n\times n$ matrix with every entry picked from $N(0,1)$, i.e., mean zero variance 1. Then, I'm wondering about $\lambda_{max}(A)$. I found Corollary 6 here (https://terrytao.wordpress.com/...
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How can we evaluate averaged products of random unitaries when the Weingarten function is singular?

In the random matrix theory literature, one often encounters identities associated with averages over ensembles of random unitaries. For a simple example let's say we're interested exclusively in $2\...
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How to understand the definition of Lebesgue measure of a Hermitian matrix?

I am reading an introduction to random matrices. In the definition of Lebesgue measure of a Hermitian matrix $$d M = \prod_{1\leq i < j\leq n} d(\Re M_{ij}) d(\Im M_{ij})\prod_{i=1}^n dM_{ii}$$ we ...
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Empirical spectral distribution, sample covariance of normal distribution

Assume I have i.i.d samples $X_i\sim\mathcal{N}(0,\Sigma)$, $1\leq i\leq n$, with an arbitrary positive definite covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. What does the spectral ...
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Is the product of two independent random matrix full rank?

The elements in matrix $A$ and $B$ are independent and absolute continous random variable. Is $AB$ be full rank with probability one? First, $A$ and $B$ must be full rank with probability one. I try ...
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Compute expected number of points in [0,1] of the sine point process

The Sine process is the determinantal point process with kernel $K(x,y)=\frac{\sin \pi(x-y)}{\pi(x-y)}$ when $x\neq y$ and $1$ when $x=y$. I am wondering how to compute the expected number of points ...
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Why $[(\mathbf{I}_N-\mathbf{A}^\top \mathbf{A})\mathbf{x}]$ is Gaussian with i.i.d. Gaussian $\mathbf{A}$?

1. Background: It is presented in the paper of approximate message passing (AMP) algorithm [Paper Link] that (the conclusion below is slightly modified without changing its original meaning): Given a ...
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How to show that $ \text{law}(\frac{1}{N}\langle \mathbf{x}_0, e^{\mathbf{J}}\mathbf{x}_0\rangle)$?

Let $\mathbf{J}=(J_{ij})_{1\le i, j\le N}$ be symmetric and let $J_{ij}$ be a centered independent random variable such that $$ E[J_{ij}^2]=1/n, \, E[J_{ii}^2]=2/n, $$ often assumed to be Gaussian ...
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Proof of the Matrix Hoeffding lemma

I am trying to find a way of convincing myself of the validity of the Matrix Hoeffding lemma. The lemma states the following: Consider a set $\{X^{(1)},\ldots,X^{(m)}\}$ of independent, random, ...
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Why is the Sine Kernel admissible as the kernel of a DPP? [closed]

The Sine DPP is given by the kernel $$K(x,y)=\frac{\sin(\pi(x-y))}{\pi(x-y)}$$ and is a well-known example of a Kernel for a determinantal point process. However, this kernel is not always positive. ...
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Limit of the smallest eigenvalue of a random Gram matrix $XX^T/n$.

Let $X\in\mathbb{R}^{n\times d}$ be a random matrix with independent rows which satisfy that $\mathbb{E}(X_iX_i^T) = I_d$ and $\mathbb{E}(X_i) = 0$. The results of P. Yaskov provide conditions on when ...
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Connection between determinantal point process kernel and Hilbert-Schmidt operator kernel

I am wondering about the connection between the kernel which gives the nth correlation function of a determinantal point process and the L^2 Hilbert space for which it uniquely defines an integral ...
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Upper bound for spectral norm of a random matrix

If we know $X$ is a $n\times n$ matrix, and each element has mean 0 and variance $b_{ij}^2$. We can also know the covariance $Cov(X_{ij},X_{kl})$. Is there any method to get the upper bound of $\...
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Ordered expected eigenvalues of Wishart matrice

Let $X \sim \mathcal{W}_p(\Sigma, n)$ follows a central Wishart distribution with scale matrix $\Sigma$ and $n$ degrees of freedom. The exact joint density function for the $p$ eigenvalues of $X$ can ...
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Another puzzling identity that arose from integrating over eigenvalues of Wishart matrices.

Let $n \ge 2$ and let $T > n $ be integers. We consider a sample covariance matrix, i.e. $c := {\bar C} \cdot Y \cdot Y^T \cdot {\bar C}^T \quad (1)$ where $Y $ is a $n \times T$ random matrix with ...
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How do we compute an integral over a unit simplex?

Let $ n \ge 2 $ and $ T > n $ be integers. The joint-distribution of eigenvalues in the Wishart ensemble subject to the underlying covariance matrix being equal to an identity matrix is given as ...
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If a random matrix converges to an invertible constant matrix $A$, does its inverse converge to $inv(A)$?

Let $B_n$ be a sequence of random matrices and $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n) = A, $$ with A invertible. Does this imply that $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n^{-...
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Powers of Non-i.i.d. Random Matrices

Suppose there is some Random Matrix $A$ which exists in $\mathbb{R}^{N\times N}$. Each element in $A$ has a different $\mu$ and $\sigma$. However, all elements are independent of eachother. $A$ is a ...
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Fixation of Random Matrix Theory on Eigenvalues

Although "fixation" carries an unintended negative connotation, why does it seem like random matrix theory is focused so intently on the eigenvalues of the matrices in question? Why is so ...
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How do I arrive at this classical/typical location of random matrix eigenvalues estimate?

Consider an $ N \times N$ Wigner matrix $H$ and define the classical or typical location of eigenvalue $\lambda_i$ as $ N \int_{\gamma_{i}}^{2} \rho(x) \text{d}x = i - 1/2 $, where $i$ indexes the ...
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Probability density function invariant under unitary transformation

Recently I have been studying Mehta's book on Random Matrices (3rd edition). In this book the author defines the Gaussian Unitary Ensemble in the set of hermitian matrices with 2 specific properties. ...
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Spectrum of randomly projected matrices

Consider a random projection of some deterministic positive $m\times m$ Hermitian matrix $A$, defined as $B:=PUAU^\dagger P$, where the $m\times m$ unitary matrix $U$ is Haar random and $P$ is some ...
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Distribution of singular values of a random matrix with columns constrained to be restrictly positive or negative.

Let $X_n$ be a $n \times n$ real matrix, and all extries expect its last column are independently sampled from an exponential distribution $exp(1)$; the entries of the last column are independently ...
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Why is the spectrum of Erdős-Rényi random graph approximately symmetric? Graphically what is symmetric spectrum?

I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős-Rényi random graph $G(n,p)$ when $np\to\infty$, and $np\to c=2,3$. Plots above is already ...
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Let $A$ have full rank almost surely and $\mathbb E[Aa] = 0 \in \mathbb R^n$. Is it true that $\mathbb E a=0$?

Let $A$ be a $m \times n$ random matrix with $m>n$ and $\operatorname{rank} A = n$ almost surely. Let $a$ be a random column vector of size $n \times 1$. Here $A$ is not necessarily independent of $...
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What can I get if I choose a vector from a given set that spans the space to maximize the determinant of rank-one perturbations of a matrix?

We have a real symmetric positive semidefinite matrix $M_0 \in \Bbb R^{d \times d}$ and a given set $K \subset \Bbb R^d$ that contains $|K|$ $d$-dimensional vectors, and it spans $\Bbb R^d$. At each ...
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Conditions for which a collection of i.i.d. random vectors has full rank almost surely

Let $X_1, \ldots, X_N:(\Omega, \mathcal F, \mathbb P) \to \Delta \subseteq \mathbb R^n$ be i.i.d. random vectors with $N>n$. Here $\Delta$ is the probability simplex. Let $$ A := \{\omega \in \...
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Invertibility of this matrix

Let $P_1$ and $P_2$ be two stochastic matrices. Prove that $I - P_1 + P_2$ is invertible. I know that the eigenvalues of $P_1$ and $P_2$ are at most 1 in magnitude. How can I handle the difference $...
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Proving this matrix limit using compactness

Let $Q = \lim_{n\to\infty} \frac{1}{n}\sum^{n-1}_{t=0}P^t$, where $P$ is a stochastic matrix. Prove that $Q$ exists and that it satisfies $QP = PQ = QQ = Q$. I have seen proofs of this using matrix ...
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Spectral norm of a random matrix whose entries are product of two standard gaussians

While solving a research problem related to approximations in neural networks, I've faced the following problem which I have not been able to solve after trying different approaches for a while. Let's ...
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Asymptotic behaviour of averaged empirical spectral distribution of normalized Wishart matrix at zero

Let $X \in \mathbb{R}^{p \times n}$ be a random matrix with independent and normally distributed elements: $X_{i,j} \sim \mathcal{N}(0, 1)$. Consider the (normalized Wishart) matrix $\hat{\Sigma} = \...
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Average number of hyperquadrants in a random subspace

Suppose I have a random $n$-dimensional linear subspace of $\mathbb{R}^m$. How many of the $2^m$ hyperquadrants does this space intersect, on average? Alternatively, what are the odds that this ...
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2 votes
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Calculating the Moments of the Marchenko-Pastur Distribution

I'm trying to follow a proof of the Marchenko-Pastur theorem. In particular I'm trying to show that the kth moment of the Marchenko-Pastur distribution: $$ a_k = \int_{(1-\sqrt{\gamma})^2}^{(1+\sqrt{\...
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Random unitary matrices distribution different from Haar random

I am trying to study a problem which requires a distribution on U(4) which is not Haar random distribution. Basically I want to have the numerical code to randomly generate $4\times4$ unitary matrices ...
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Asymptotic convergence of shifted and scaled Wigner matrix

Let $\frac{X}{\sqrt n}$ be a Wigner matrix, such that $X_{ij}$ are iid random variables, with mean 0 and variance 1, with $X_{ij} = \overline{X_{ij}}$ for $i > j$. Then we know by Wigner's Theorem ...
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Problem with Wigner's Surmise

My numerical calculation does not agree with Wigner's Surmise. On page 11 of https://arxiv.org/pdf/1712.07903.pdf, it derives a distribution for the spacing between the eigenvalues of a matrix. I ...
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2 votes
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$\ell_1$ norm of random projection

Let $G_{m, n}$ denote the Grassmannian manifold, i.e. the set containing all possible subspaces of $R^m$ with dimension $n$. Let $E \in G_{m, n}$. We can associate with $E$ an orthogonal projection ...
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1 vote
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Recommendation of an Introduction to Integrable Probability

I am interested in the topic of Integrable Probability with topics like KPZ Universality and Tracy-Widom Distribution. What are the prerequisites for this subject? I have standard Undergraduate level ...
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3 votes
1 answer
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What is the probability that $Ax = y$ for a random matrix $A$?

Let $m, q,$ and $n$ be integers and $m$ be much greater than $n$. Let $q$ also be a prime number. Consider an $m \times n$ matrix $A$. Each entry of $A$ is chosen uniformly at random from $\mathbb{Z}...
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Uniform sampling on the set of symmetric positive-semidefinite matrices with bounded entries

What is the most correct way to randomly generate a (square) symmetric positive-definite matrix $A$ with nonnegative entries bounded in [0,1]? One way I can think of is by sampling a matrix $X$ from ...
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How does one denote the set of all positive-definite square matrices? [duplicate]

For example, can I write: The matrix $X \in \mathbb{R}^{p \times p}_{>0}$ follows a Wishart distribution $$ X \sim \mathcal{W}(V,n) $$ where $\mathbb{R}^{p \times p}_{>0}$ is the set of all ...
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Bound on denominator of echelon form of integer random matrix

Let $M$ be an $\ell\times k$ matrix, with $k>\ell$ and $\ell\ge1$. The entries of $M$ are integers independently and uniformly randomly chosen between $-n$ and $n$. Excluding degenerate cases, the ...
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3 votes
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Probability of dense subgraph in a random graph

What is the probability that a random graph with $n$ vertices and degree sequence $\left(d_i\right)_{i=1..n}$ has a subgraph of $k$ vertices and density $\delta$? The random graph is typically ...
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1 vote
1 answer
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Minimal eigenvalue of symmetric random matrix generated by a random vector

Suppose we have a random row vector $V_n=(v_1,...,v_n)$, where $v_1,...,v_n$ are iid and real-valued. We now create the matrix $M_n=\frac{1}{n}V^TV$. Are there any nontrivial assumptions on the ...
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Crossing out an orthogonal-valued function out of expected value of product

Main part: Let $G$ be a locally compact group with corresponding Haar measure $\mu$. Let $f, g: \Omega \to G$ be arbitrary measurable functions. It would be really nice to have: $$\mathbb{E} [gf] = \...
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Some convergence relation with a summation of a Wigner matrix and its resolvent

Given a Wigner random matrix $X \in \mathbb R^{N\times N}$ ($X_{j,i} = X_{i,j}$ and $X_{i,j} \sim \mathcal N(0,1)$), and $G(z) = (\frac{X}{\sqrt N} - zI_N)^{-1}$ its resolvent, there seems to be that: ...
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1 vote
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Deviation of Random Projections

Let $P$ be an orthogonal projection in $\mathbb{R}^n$ onto an $m-$dimensional random subspace uniformly distributed in the Grassmannian $G_{n,m}$. Let $T$ be a bounded subset of $\mathbb{R}^n$. Let $x$...
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