Questions tagged [random-matrices]
For questions concerning random matrices.
736
questions
0
votes
1
answer
24
views
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 $y$ ...
- 2,226
0
votes
0
answers
9
views
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 ...
- 155
4
votes
2
answers
66
views
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 ...
- 523
0
votes
0
answers
15
views
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/...
- 526
0
votes
0
answers
12
views
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\...
- 165
0
votes
0
answers
15
views
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 ...
- 1,387
0
votes
0
answers
18
views
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 ...
- 7,304
1
vote
0
answers
11
views
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 ...
- 11
0
votes
1
answer
23
views
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 ...
- 1,294
4
votes
1
answer
168
views
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 ...
- 160
0
votes
0
answers
14
views
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 ...
- 1,808
1
vote
1
answer
59
views
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, ...
- 170
1
vote
1
answer
32
views
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. ...
- 1,294
0
votes
0
answers
23
views
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 ...
- 33
2
votes
1
answer
38
views
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 ...
- 1,294
0
votes
0
answers
30
views
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 $\...
- 399
0
votes
0
answers
23
views
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 ...
- 107
1
vote
0
answers
21
views
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 ...
- 9,043
1
vote
1
answer
34
views
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 ...
- 9,043
1
vote
0
answers
21
views
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^{-...
- 554
2
votes
0
answers
33
views
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 ...
- 21
0
votes
0
answers
19
views
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 ...
- 21
0
votes
0
answers
5
views
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 ...
- 425
0
votes
0
answers
18
views
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. ...
- 54
0
votes
1
answer
38
views
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 ...
- 247
0
votes
1
answer
59
views
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 ...
- 23
2
votes
0
answers
32
views
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 ...
- 927
1
vote
1
answer
28
views
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 $...
- 1,153
1
vote
0
answers
71
views
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 ...
- 11
0
votes
0
answers
18
views
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 \...
- 1,153
3
votes
1
answer
66
views
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 $...
- 503
0
votes
0
answers
50
views
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 ...
- 149
1
vote
0
answers
76
views
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 ...
- 55
0
votes
0
answers
22
views
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} = \...
- 56
1
vote
0
answers
29
views
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 ...
- 1,537
2
votes
1
answer
44
views
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{\...
- 194
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 ...
- 21
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 ...
- 47
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 ...
- 25
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 ...
- 925
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}...
- 523
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 ...
- 63
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) $$
where $\mathbb{R}^{p \times p}_{>0}$ is the set of all ...
- 92
0
votes
0
answers
7
views
Bound on denominator of echelon form of integer random matrix
Let $M$ be an $\ell\times k$ matrix, with $k>\ell$ and $\ell\ge1$. The entries of $M$ are integers independently and uniformly randomly chosen between $-n$ and $n$.
Excluding degenerate cases, the ...
- 1,997
3
votes
0
answers
59
views
Probability of dense subgraph in a random graph
What is the probability that a random graph with $n$ vertices and degree sequence $\left(d_i\right)_{i=1..n}$ has a subgraph of $k$ vertices and density $\delta$?
The random graph is typically ...
- 703
1
vote
1
answer
29
views
Minimal eigenvalue of symmetric random matrix generated by a random vector
Suppose we have a random row vector $V_n=(v_1,...,v_n)$, where $v_1,...,v_n$ are iid and real-valued.
We now create the matrix $M_n=\frac{1}{n}V^TV$.
Are there any nontrivial assumptions on the ...
user934432
0
votes
0
answers
11
views
Crossing out an orthogonal-valued function out of expected value of product
Main part:
Let $G$ be a locally compact group with corresponding Haar measure $\mu$. Let $f, g: \Omega \to G$ be arbitrary measurable functions. It would be really nice to have:
$$\mathbb{E} [gf] = \...
- 669
0
votes
0
answers
18
views
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:
...
- 75
1
vote
0
answers
31
views
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 $x$...
- 533