Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
871 questions
2
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A naive question about non-Hermitian random matrix
The resolvent of a matrix $\mathbf{A}$ is defined as
\begin{equation}
\mathbf{G}_{\mathbf{A}}(z) = \left(\mathbf{A} - z \mathbf{1}_n\right)^{-1}, \quad z \in \mathbb{C} \setminus \sigma(\mathbf{A}),
\...
- 109
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votes
0
answers
57
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
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2
votes
0
answers
100
views
Stiefel C-manifold: volume and uniform distribution
I'm trying to define the uniform distribution on the Stiefel $C$-manifold (Downs 1972), given by $\mathcal{V}_{p,n}(C) = \{ X \in \mathbb{R}^{n \times p} : X' X = C \}$ for $n \geq p$ and $C > 0$. ...
- 21
4
votes
1
answer
244
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Distinct eigenvalues of random matrix over finite field
Let $A$ be a uniformly random matrix in $\mathrm{M}_n(\mathbf{F}_p)$.
It is well known that, as $p$ is fixed and $n$ tends to infinity, $A$ has repeated eigenvalues (over the algebraic closure $\...
- 309
1
vote
0
answers
40
views
Asymptotic unitary invariance of rank-one spiked Gaussian matrix
I'm working on some Random Matrix Theory related stuff for my thesis, and i've come across the following problem:
Consider a (normalized) spiked Wigner matrix $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{...
- 11
1
vote
0
answers
74
views
Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure
$$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$
for ...
- 131
3
votes
1
answer
98
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Error bound for MonteCarlo estimate of elements in Gram-Matrix
Suppose I have a $n\times n$-symmetric positive-definite matrix $A$ with elements:
\begin{align}
[A]_{ij}=\int_{\Omega}f_i(x)f_j(x) \, dx, \quad i,j=1,\ldots,n
\end{align}
where $\Omega\subset \mathbb{...
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1
vote
1
answer
48
views
Parameters of Wishart distribution and generalized inverse
I recently came across the Wishart Distribution and a few things are unclear to me.
The Wikipedia page for the Wishart Distribution says that if $G=[g_1 \vert \; g_2\vert \; \ldots \vert g_n]$ is a $...
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votes
2
answers
135
views
Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 150
7
votes
1
answer
274
views
Smallest eigenvalue of a random matrix
Let $A \in \mathbb R^{n\times n}$ be a positive semi-definite matrix,
and let $b \in \mathbb R^n$. For a random vector $x \sim \mathcal N(0, I_{n\times n})$, consider the random matrices
$$
B_1 = A + ...
- 211
2
votes
0
answers
68
views
Nonlinear random matrix equations
Let $C$ be a matrix; $v$ be a column vector;
$P$, $\Delta$ are random matrices;
$x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
0
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0
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30
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Analytic / algebraic characterization of the limiting value of the unique nonnegative root of a polynomial
I'm interested in the following problem which arises from some "random matrix theory" calculations. Let $\phi,s_1,s_2, p > 0$ with $p \in [0,1]$, and set $p_1=p$, $p_2=1-p$, and $q_k := ...
- 6,853
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Why has the random Koopman matrix $ G_{xx}^{(-)} G_{yx} $ only eigenvalues on the complex unit circle?
Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution,
e.g.
$$ U_{i j} \sim N(0,1) \quad \quad i,j=1,...n+1$$
Let $ G=U U^* $ be a Gram matrix where $ U^* ...
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43
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Variance and expected value of power of normal matrix elements
Here is the next problem: we have $n\times n$ random matrix $\hat{A}$, each element of this matrix is independent real random variable with normal distribution ($\mu=0, \sigma^2=1/n$). Matrix is no-...
0
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0
answers
27
views
Expectation of the Inverse of Sum of Weighted Identity Wishart Matrices
As noted in the title, suppose $A=\sum_{i=1}^{n}\lambda_ig_ig_i^T$ where $g_i\sim N(0,I_d)$ and $d < n$.
Essentially, $A$ is the weighted sum of $W_d(I_d,n)$. Ultimately, I want to calculate the ...
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1
vote
0
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80
views
Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
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98
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Bounding the norm of a sum of fourth-order Gaussian vectors
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$, where $\Sigma = \operatorname{diag}(\lambda_1, \cdots, \lambda_k)$ and
$\lambda_1 \ge \cdots \ge \...
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answers
203
views
Spectrum of $\prod_i^d \left(I-x_ix_i^T\right)$ for isotropic $x_i$
Suppose $x_i\in \mathbb{R}^d$ are IID isotropic random vectors with $\|x_i\|=1$ and matrix $A_d$ is defined as follows:
$$A_d=\prod_i^d \left(I-x_ix_i^T\right)$$
Is anything known about the spectrum ...
- 2,442
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vote
0
answers
43
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Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
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3
votes
0
answers
131
views
Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
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2
votes
1
answer
111
views
"High complexity" of eigenbasis of Wigner matrices?
Let $W$ be an $N \times N$ complex Wigner matrix, i.e. i.i.d. entries restricted to Hermitian matrices. Let $W=UDU^{\ast}$, i.e. $U$ encodes the eigenbasis of $W$. Are there any statements known about ...
- 3,323
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votes
0
answers
29
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Spectral bound for sample covariance matrix without assuming $X = \Sigma^{\frac{1}{2}} Z$
Let $X$ be a random $(p \times n)$-matrix with iid centered columns and suppose the entries of $X$ all have light tails (in a strong enough sense, for example sub-Gaussian). Are there any results ...
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0
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29
views
What is known about the distribution of eigenvectors for random positive semidefinite matrices?
Let $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$.
Define the random $d \times d$ real positive semidefinite matrix,
$$
S_n = \frac{1}{n} \sum_{i=1}^n ...
- 390
1
vote
1
answer
274
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Does free multiplicative convolution become free additive convolution under logarithm?
Let $X$ and $Y$ be two $n\times n$ random matrices that have zero measure over degenerate matrices. For a positive definite matrix with eigen-decomposition $A = U \Lambda U^\top$, let $\log(A) = U \...
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votes
0
answers
21
views
Bound p-th order moments for a random Wishart matrix to show the sub-exponential property
Let $a\in\mathbb{R}^k$ be a random vector sampled from $N(0,\Sigma)$. Let $X = aa^T - \Sigma$. Then we have $\mathbb{E} X = 0$. Can we find a constant $C\in\mathbb{R}$ and another fixed matrix $A\in\...
- 97
1
vote
1
answer
111
views
Matrix concentration inequality for unbounded (sub-exponential) matrices
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...
- 97
8
votes
1
answer
428
views
Wishart matrices: are eigenvalues and eigenvectors independent?
Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.
In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
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1
vote
1
answer
84
views
Limiting value of Stieltjes transform of sum of independent Wishart matrices
Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
- 6,853
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votes
2
answers
192
views
Behavior of a Wishart quadratic form
Let $X \in \mathbb{R}^{n \times d}$ be a random matrix with iid standard Gaussian entries. Let $e_1$ denote the first canonical basis vector in $\mathbb{R}^d$. Define
$$
P_d(\lambda) = (1-\lambda) e_1 ...
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3
votes
1
answer
145
views
Orthogonal projection $X X^+$ from random Gaussian matrix $X$
Given a standard Gaussian matrix $X\in\mathbb{R}^{n\times d}$, $d<n$, with entries sampled i.i.d. from $\mathcal{N}(0,1)$, is the corresponding orthogonal projection $X X^+ = X (X^\top X)^{-1} X^\...
2
votes
1
answer
246
views
Inequality with Hermite polynomials
Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...
- 43.2k
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votes
1
answer
159
views
Dot product of a randomly orientated vector and a fixed vector
Let us consider a random variable $Z$ with a probability density function $f$ with respect to the Haar measure on $\mathrm{SO}(3)$. Next, we consider two fixed normal vectors $u,v$ in $\mathbb{R}^3$. ...
user528399
0
votes
1
answer
108
views
RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
- 6,853
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votes
1
answer
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Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?
I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors.
Exercise 3.1.7 from the book is the following.
Let $X =...
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votes
0
answers
116
views
Random matrix with power law decay in eigenvalues
What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact.
I want to find random matrix ensembles that naturally ...
- 433
6
votes
1
answer
252
views
Poisson kernel for the orthogonal groups
For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
- 1,549
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votes
0
answers
56
views
Random pseudo-inverse matrix problem
Given a matrix $M \in \mathbb{R}^{n \times N_d}$, $N_d \gg n$ and $\mathrm{rank}(M) = n$, the entries of $M$ are denoted as $M_{[ij]}, i = 1,...,n, j = 1,..., N_d$ and $M_{[ij]} \in [-\textbf{m}, \...
1
vote
1
answer
110
views
Bounded density for determinant of GOE
Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
- 1,352
1
vote
1
answer
99
views
Maximum column norm of random $A^{-1}B$
Suppose that $A$ is an $n$ by $n$ Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let $b$ be a $n$-Gaussian vector. Then it could be easily proven that the ...
- 33
1
vote
0
answers
92
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
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votes
2
answers
752
views
Computing Haar measure of matrices sampled from SO(n)
I am looking to sample uniform matrices from SO(n).
I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
- 181
1
vote
2
answers
66
views
Distribution of the constraint matrix conditioned on the solution of the linear system
Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
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0
votes
0
answers
35
views
Question about the spectrum of a deformed GOE matrix
Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the ...
- 1
0
votes
1
answer
68
views
Convergence of edge eigenvalues for Gaussian matrices
I am reading this lecture note.
I have a difficulty in understanding the third section in chapter 6. Particularly, in Theorem 4.1, they claimed that
Let $X$ be a Gaussian Wigner matrix satisfying ...
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2
votes
1
answer
124
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Functional calculation for Hermitian matrices
First, let me recall some useful definitions
We recall that if $A=U \Lambda U^*$ is a Hermitian matrix with $U U^*=U^* U=I$ and $\Lambda=\operatorname{diag}\left(\lambda_1\right)$ and $f: \mathbb{R} \...
- 129
0
votes
0
answers
92
views
Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
- 503
0
votes
0
answers
42
views
Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
0
votes
0
answers
48
views
Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
- 1,755
2
votes
1
answer
128
views
Eigenvalue analysis of $X^T (XX^T + \mathrm{Id})^{-1} X$ for $X$ iid random matrix
Consider the following quantity
$$X^T (XX^T + \mathrm{Id})^{-1} X,$$
where $X \in \mathbb{R}^{m\times n}$ is a iid random matrix with 0 mean and finite variance.
The empiric covariance matrix ${X^T X}$...
- 2,306
0
votes
0
answers
77
views
Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?
Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function
$$
f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big],
$$
over the class of real ...
- 390