# 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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### 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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### 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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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{\... 0 votes 0 answers 11 views ### 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 ... 1 vote 0 answers 15 views ### 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 ... 0 votes 1 answer 57 views ### 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 ... 2 votes 0 answers 66 views ### \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 ... 1 vote 0 answers 61 views ### 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 ... 3 votes 1 answer 92 views ### 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}... 0 votes 0 answers 14 views ### 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 ... 0 votes 0 answers 20 views ### 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) $$where \mathbb{R}^{p \times p}_{>0} is the set of all ... 0 votes 0 answers 7 views ### Bound on denominator of echelon form of integer random matrix Let M be an \ell\times k matrix, with k>\ell and \ell\ge1. The entries of M are integers independently and uniformly randomly chosen between -n and n. Excluding degenerate cases, the ... 3 votes 0 answers 59 views ### Probability of dense subgraph in a random graph What is the probability that a random graph with n vertices and degree sequence \left(d_i\right)_{i=1..n} has a subgraph of k vertices and density \delta? The random graph is typically ... 1 vote 1 answer 29 views ### Minimal eigenvalue of symmetric random matrix generated by a random vector Suppose we have a random row vector V_n=(v_1,...,v_n), where v_1,...,v_n are iid and real-valued. We now create the matrix M_n=\frac{1}{n}V^TV. Are there any nontrivial assumptions on the ... 0 votes 0 answers 11 views ### Crossing out an orthogonal-valued function out of expected value of product Main part: Let G be a locally compact group with corresponding Haar measure \mu. Let f, g: \Omega \to G be arbitrary measurable functions. It would be really nice to have:$$\mathbb{E} [gf] = \...
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: ...
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 $x$...