Questions tagged [random-matrices]
For questions concerning random matrices.
831
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Probability of row sum zero random matrix being PSD
Consider $A$ to be a symmetric random matrix. When $i<j$, $A_{ij}=1+\sigma W_{ij}$ where $W_{ij}$ are standard gaussian variables, $A_{ij}=A_{ji}$ and $A_{ii}=\sum_jA_{ij}$. The $\sigma$ is a ...
- 139
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Universality of correlators in $\beta$ Hermite ensembles
Recently I got interested in the world of Random Matrix Models and I bumped into some generalizations of the usual random matrix theories classified by Dyson whose probability density functions are:
$...
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0
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26
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Entropy increase of a random walk on the sphere
Let $X_0$ be some vector on the unit sphere of $\mathbb{R}^n$ and let $\rho_0$ be the distribution of its entries,
$$
\rho_0(z) = \frac{1}{n} \sum_i \delta\left(z-X_0^{i}\right)
$$
Assume that $n$ is ...
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24
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Generalization of Marchenko-Pastur law [closed]
I have a question about the random matrix theory. Say $X \in R^{M \times N}$ is a random matrix such that the each row of $\sqrt{N} X$ are independently complex random variables with mean zero, ...
- 85
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33
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Joint Probability Distribution between columns of Haar Random Unitary
Let U be a unitary matrix sampled from the Haar measure over U(D). We can decompose U as:
\begin{equation}
U = (U_{1}, \cdots , U_{D})
\end{equation}
where the $i^{th}$ column is expressed as a ...
- 41
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0
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14
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Eigenvalues of multivariate t-wishart and wishart matrix
If $W$ is a wishart matrix with Identity Covariance and $n$ degrees of freedom. And another matrix $X= v*W*S$ where $S$ is a diagonal matrix with diagonal elements as iid inverse-$\chi^2$ distributed ...
- 33
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10
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Bounding the distance between projection onto random subspace and projection onto the average subspace
Using results like the Davis-Kahan theorem and Wedin's $\sin \theta$ theorem, we can bound quantities like $\|\widehat{V}\widehat{V}^{\top} - VV^{\top}\|_F$ where $X = V\Sigma V^{\top}$ and $\widehat{...
- 471
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22
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Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns
Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i....
- 31
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Free cumulants of Gaussian matrices with independent entries
Setup: Remark 5.1 in this work states that when $X\in\mathbb{R}^{n\times p}$ have i.i.d. $N(0,1/p)$ entries, the limit distribution of $XX^\top$ is the Marcenko-Pastur law, with limiting rectangular ...
- 694
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21
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Expected Value Largest Eigenvalue of a Random matrix
For a symmetric matrix whose entries are chosen uniformly at random $[-1,1]$ how do I find the expected value of the maximum eigenvalue of the matrix. As far as I understand finding this expected ...
- 113
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What is the expected number of walks with length 𝑘 in Erdős–Rényi random graph?
Let $G(N, p)$ be a directed Erdős–Rényi random graph with edge probability $p$. Let $W_k$ denote the number of walks (potentially with repeated vertices, or repeated edges) of length $k$ beginning at ...
- 58
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1
answer
50
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Eigenvalue density of a matrix whose kth moment is equal to 1/k
This is an exercise from the book "A First Course in Random Matrix Theory" by Potters and Bouchaud.
We're interested in finding the eigenvalue density $\rho(\lambda)$ of a large random ...
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37
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Why are square Bernoulli matrices invertible with high probability?
Given a matrix $A\in\mathbb{R}^{m\times n}$ with entries of $A$ being sampled i.i.d. from $\text{Bernoulli}(\alpha)$, where $\alpha\in(0,1)$ is a fixed constant. This paper (2nd sentence below ...
- 694
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Sum of (non-free) symmetric and asymmetric random matrices
Consider random matrices $X,Y\in \mathbb{R}^{N\times P}$, i.e., $X_{ik}\sim{\cal N}(0,1/N)$ i.i.d. and likewise for $Y$.
The eigenvalues of the symmetric covariance matrix $X^TX$ reside on the real ...
- 413
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17
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A question in operator norm of random matrix by epsilon net in Tao's RMT notes
I am reading Tao's Random matrix notes, specifically Proof of Corollary 2.3.5.
Corollary 2.3.5. (Upper tail estimate for iid ensembles). Suppose that the coefficients $\sigma_{ij}$ of $M$ are ...
- 139
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SVD distribution of linearly transformed Gaussian ensemble
The joint pdf of the singular values of the $m \times n$ Gaussian ensemble $X = x_{i,j} $, where the $x_{i,j} $'s are independent Normal(0,1) samples and the eigenvalues of the associated Wishart ...
- 11
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33
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Matrix Vector multiplication of Gaussian Matrix and Gaussian Vector
I am curious about how $y$ is distributed if:
$$y=Ax,$$
with Gaussian Matrix $A \in \mathbb{R}^{m \times n}$, every entry of the matrix $e_{ij} \sim N(0, 1)$ and Gaussian Vector $x \in R^n, \mathbb E(...
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Definition of Haar orthogonal matrices.
I recently came across the terminology "Haar orthogonal matrices" in this paper (page 4), where they stated that it is the matrix that comes out from singular value decomposition. However, I ...
- 694
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Can we generalize the idea of spatially coupled gaussian matrices to rotationally invariant matrices?
Setup: An $(\omega,\Lambda)$ base matrix $W\in\mathbb{R}^{R\times C}$ is described by the coupling width $\omega\geq1$ and the coupling length $\Lambda\geq2\omega-1$. The matrix has $R=\Lambda+\omega-...
- 694
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13
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Smallest Singular Value of submatrices from a column-orthogonal matrix
Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\...
- 13
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1
answer
73
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Spectral density of non-Hermitian random matrix
I have a question regarding the computation of the spectral density of some non-Hermitian random $N\times N$ matrix $A$. Following Rogers and Castillo, 2009, we denote the collection of eigenvalues of ...
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36
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Matrix Expectation Involving $6^\text{th}$ Degree Gaussian Moments
Problem. I would like to compute the following expectation:
$$F(A, B) = \mathbb{E}_X\left[X X^T A X X^T B X X^T \right]$$
where $X$ is a $d\times n$ matrix with every entry being i.i.d. standard ...
- 321
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19
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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 \...
- 834
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51
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Properties of unitary transformations of random matrices
I'm trying to find an upper bound for the following expression
$$
\mathbb{E} || \Psi_2^{+} \Psi_1 A ||_F
$$
Where $A \in \mathbb{R}^{(n-k) \times m}$ is fixed, and the matrices $\Psi_1 \in \mathbb{R}^{...
- 951
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1
answer
31
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Expected Smallest Eigenvalue of a Random Matrix whose Expectation is Positive Definite
Suppose $M$ is a random symmetric $n\times n$ matrix satisfying $\lambda_{\min}(\mathbb{E}M)>0$, where $\lambda_{\min}$ denotes the smallest eigenvalue.
My question: What can we say about the ...
- 629
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Showing the upper bound of the operator norm of a rectangular matrix
Question: Given a square random matrix $X\in\mathbb{R}^{p\times p}$ where
$$
X_{ij}\stackrel{iid}{\sim}\text{Bernoulli}(\alpha),
$$
where $\alpha\in(0,1)$, satisfies the condition
$$
\|X\|_{\text{op}}&...
- 694
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22
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Going from S-transform to eigenvalue density?
Suppose we have a random matrix variable with the S-Transform below with $n>=1$
$$g(z)=\frac{1}{(1+z)^n}$$
What is the eigenvalue density around 0?
For $n=1$, this corresponds to Marchenko-Pastur ...
- 4,739
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12
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What is the operator norm of a function of n multiply a gaussian matrix?
Let $W$ be Gaussian Wigner matrix, and it is well-known that its operator norm satisfies
$$P(\|W\|\leq 2\sqrt{n}+t\geq 1-2\exp(-ct^2)).$$
My question is too basic but I cannot figure it out:
What is ...
- 139
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17
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Probability of a row in a permutation matrix being "correct"
Question: We have a uniformly sampled permutation matrix $\Pi\in\{0,1\}^{n\times n}$. What is the probability that the $i$th row in $\Pi$ is in the "correct" position, which means that the $...
- 694
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impose condition on random matrix such that its element sum and row sum satisfies certain condition
Assume $a_{ij}:=1+ c_{ij}$ is element of $n\times n$ symmetric matrices $A$ with diagonal element equals to $0$.
What condition we can impose on matrix $C$ (whose elements are $c_{ij}$), such that
$$...
- 139
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Statistical Properties of Eigenvalues
While playing around with some numpy and a random number generator, I found this odd bit. I set up a function to generate a matrix of some size and fill it with pseudorandom integers. Then, using ...
- 83
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Properties of a random permutation matrix
Given a uniformly sampled permutation matrix $\Pi\in\{0,1\}^{n\times n}$, what can we say about the matrix $E$ where
$$
\Pi = I + E,
$$
where $I$ is the identity matrix. More precisely. what can we ...
- 694
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Singular values of random matrix with IID entries
Chart below is log-log plot of empirical CDF of the distribution of singular values for a matrix with IID random entries.
It tends to $O(x)$ for small $x$ regardless of the distribution, why?
...
- 4,739
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An integral involving products of random matrices
I came across an integral involving a product of random matrices that I'm unable to evaluate, and I'm curious about whether or not there's a known solution. For context, I started with the following ...
- 43
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Explicit expressions for the integrals of second degree polynomials in the orthogonal group
Wikipedia has explicit formulas for first and second degree polynomial integrals in the unitary group using Weingarten functions (https://en.wikipedia.org/wiki/Weingarten_function).
Is there any ...
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Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$
This is a call for help to the random-matrix-theory savvy people. I've observed the below equality in experiments, and have been looking for a proof in the RMT literature but couldn't find one. I'd ...
- 129
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Concentration of inverse sample covariance matrix
Let $X_1, X_2, \dots, X_n \sim X$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\mathbb{E}[XX^{\top}] = \Lambda$. Let $\hat{\Lambda} = \sum_{j = 1}^{n} X_j X_j^{\top}$ be the empirical covariance ...
- 2,260
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Some hint on Generalization or Extension of the convergence related to random matrix?
I'm dealing with a problem that I never met before, in which the question statement is clean & brief, but not sure about the skills/theorem that I should rely on, may be LLN, CLT; or maybe other ...
- 1
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answer
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A closed form expression for the marginal probability density of eigenvalues of real Wishart matrices
This is a follow-up to my other question here . Define ${\mathfrak a}:= (T-N-1)/2 \in {\mathbb N}$ for $T > N+1$ and both $N,T \in {\mathbb N} $. Go to the link for the definition of the prefactor $...
- 10.8k
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Asking if a diagonal Gaussian matrix satisfies a compressed sensing assumption
This is regarding the compressed sensing problem. In the problem $y=X\beta$ where $y\in\mathbb{R}^{n}$, $X\in\mathbb{R}^{n\times p}$, and $\beta\in\mathbb{R}^{p}$, it is known that a random matrix $X$ ...
- 694
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Asking if a Bernoulli matrix and its variants satisfy the Restricted Isoperimetric Property (RIP) condition
This is regarding the compressed sensing problem that I recently learned about. In the problem $y=X\beta$ where $y\in\mathbb{R}^{n}$, $X\in\mathbb{R}^{n\times p}$, and $\beta\in\mathbb{R}^{p}$, it is ...
- 694
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[Follow up question]: What is the corresponding distribution of the SVD assuming U and V matrices are uniformly sampled wrt Haar measure?
I am asking for a follow up discussion to a question given by another user, based on the following URL link from this website asked about a year ago:
https://mathoverflow.net/a/439140/129496
In ...
- 882
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6
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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><o_i|...
- 33
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A set of identities involving the Gaussian hypergeometric function and its generalizations to several variables.
In my other question we derived a "closed form expression" for the spectral moments of Wishart random matrices (under the assumption that the underlying correlation matrix is an identity). ...
- 10.8k
2
votes
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39
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What's the limiting distribution of the eigenvalue and eigenvectors of the sample covariance matrix?
The question is from the book 'Statistical Models and Methods for Financial Markets', Page 43.
Suppose $\boldsymbol{x_1,..., x_n}$ are $n$ independent observations from a multivariate population with ...
- 21
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vote
1
answer
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What is the probability a random orthonormal matrix will be a rotation matrix?
Construct an orthonormal matrix in $\Bbb{R}^n$ in the following manner:
Choose a point uniformly on the unit sphere in $n$ dimensional space (can be done by sampling from an $n$ dimensional isotropic ...
- 6,463
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25
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matrix bernstein's inequality: from tail probability to expectation
Let $X_i$ be independent, mean zero, $n\times n$, symmetric random matrices. $\|X_i\|\leq K$ almost sure for $\forall I$.
We have matrix Bernstein's inequality for the tail probability as follows
$$\...
- 139
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vote
1
answer
36
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A closed form expression for spectral moments of arbitrary order in the Wishart ensemble
We are interested in calculating the spectral moments of random matrices sampled from the Wishart distribution.
Let $N,T$ be positive integers with $T> N$ .
Then the quantities in question are ...
- 10.8k
0
votes
2
answers
29
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Eigenvalues of skew-symmetric matrices from 2D random points
I am generating $n$ random points in two dimensions $(x_i, y_i)$. Then I form this skew-symmetric matrix
$$ M_{ij} = \begin{cases} x_i y_j - x_j y_i & \text{if } i < j \\ 0 & \text{if } i=...
- 453
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0
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21
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Expected norm-squared of one random vector projected onto others, all iid
For $k\leq n$, $x_1,\dots,x_k \in \mathbb{C}^n$ are independent identically distributed random vectors almost surely unit norm and with span dimension $k$. Call $X=[x_2,\dots,x_k]$. I am studying $\...
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