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Questions tagged [random-matrices]

For questions concerning random matrices.

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems may be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

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Expected power of a determinant of a random matrix

Say $X_{ij}$ are i.i.d. random variables with known moments $\mathbb{E}\left[X_{ij}^n\right] = \mu_n$. Given a random matrix $A = \{X_{ij}\} _{n\times n}$, what is the expected value $$V_m = \mathbb{E}\left[(\det A)^m\right]$$ equal to? (Where $m$…
Machinato
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Spectral Radius of Random Matrix with Column-wise Variance

Consider a random matrix with mean-zero, independent elements with variance defined column-wise: $M_{ij}$ with $$\mathbb{E}[M_{ij}^2]=\frac{\sigma_j^2}{N}$$ and assume that the average the variances is $\sigma^2$, say via a doubly-stochastic process…
user3371057
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Assigning students in groups

I have a project with 60 students and 6 teachers. I'd like to know if I can use Sage or Mathematica to assign those 60 students in 6 groups. There would be 6 sessions (a month each). Each student has to work once with each teacher and avoid being in…
Kaiserfilk
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How to compute the covariance of a random matrix.

I want to know the best way of computing the covariance matrix of a random matrix $M$. Assume I have a $p$ by $p$ random matrix $S=\frac{1}{n}\sum_{t=1}^nV_tV'_t$ where $V_t$ are i.i.d $p$-random vectors drown from $N(0,W)$. Define…
O.S
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Unitary matrices and Riemann zeros is there an error?

when they exploit the relationship by Berry and Keating between the Riemann zeros and eigenvalues of random matrices why do they choose $$ \frac{\gamma _{n}}{2\pi}log \frac{\gamma}{2\pi e} $$ as a Random variable however why are they ignoring the…
Jose Garcia
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Expectation of $XX.T$ vs Expectation of X * Expectation of X.T

I need to understand what follows: Given $X \in \mathbb{R}^{T \times n_0}$ $W \in \mathbb{R}^{n_0 \times n} : W_{ij}\sim \mathbb{N(0, \frac{1}{n_0})} \space i.i.d.$ $\sigma : \mathbb{R} \rightarrow \mathbb{R} $ so that $\sigma(x) = max(0,x)$ $S =…
fabianod
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Does the Marchenko-Pastur Law imply small singular values when p ~= n?

I am reading about the Marchenko-Pastur Law (https://people.math.wisc.edu/~valko/courses/833/2009f/lec_6_7.pdf) and trying to decipher the main theorem so it is more intituitive. Would it mean in any case that if $p/n \rightarrow 1$ (or $y$ is close…
kloop
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Multiplication (left or right) of Ginibre Orthogonal (GiOE) or Unitary (GiUE) with a Haar-random unitary matrix

Assume $G$ is a member of the GiOE (GiUE). I sample a Haar-random unitary matrix $U$ and multiple the two, either like $G \cdot U$ or $U \cdot G$. Would it be correct to say that the GiOE (GiUE) is invariant under this operation?
trurl
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Definition of GOE matrix

A matrix $A_N$ is called Gaussian orthogonal ensemble (GOE) if $A_N$ is symmetric; $A_N(i, i) \sim \mathcal{N}(0,2)$ for $i=1, \dots, N$ and $A_N(i, j) \sim \mathcal{N}(0,1)$ for $1 \leq i < j \leq N$; $(A_N(i,j))$ independent for $1 \leq i \leq j…
fpmoo
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Moments of fixed observable in the eigenbasis of a Gaussian Orthogonal Ensemble

I need a clear derivation of the moments of a fixed observable $\hat{O}$ in the eigenbasis of a Gaussian Orthogonal Ensemble or a Gaussian Unitary Ensemble. Write $\hat{O} = \sum_i o_i |o_i>
Snpr_Physics
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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 have the product of measures of the real and complex…
Charlie Chang
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Optimal squares to choose in the football square game

The game goes like this: Before the (American) football game begins, $5$ players each choose $20$ squares on a $10 \times 10$ grid. After choosing, the numbers $0-9$ are chosen at random twice, once to label each row and once to label each column.…
Ty Jensen
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Can I expand the results of the product of independent vectors to the product of correlated vector in Random Matrix Theory?

Consider I have an $M\times 1$ vector $\boldsymbol x$ whose elements are i.i.d. random variables. Based on Theorem 3.4 in [1], as $M\to\infty$, we have : $\boldsymbol{x}^H\boldsymbol{A}\boldsymbol{x}\xrightarrow{a.s.}tr(\boldsymbol{A})$ Now,…
Masoud
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identification of random-matrices

May any colleague help us with the following problem. We have encountered an infinite matrix (occupied by positive integers and zeros). The column space of the matrix could be in parts interpreted as sequences but not sure if there is an overall…
al-Hwarizmi
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Are these random matrix processes equivalent?

Suppose I would like to randomly fill an empty matrix (consisting of zeroes) with a particular number of each of the elements $a$ and $b$. One way is first fill the matrix with the proper number of $a$'s and then with the proper number of $b$'s.…
user519413
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