Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
900 questions
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Convex concentration of a tensor-squared spherical vector
Let $X$ be a random vector in $\mathbb{R}^n$. We say that $X$ has the convex concentration property with constant $K > 0$ if for every $1$-Lipschitz convex function $\varphi : \mathbb{R}^n \to \...
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Change in spectrum after rank-1 update to a large random matrix
I am trying to learn random matrix theory myself from the book https://zhenyu-liao.github.io/book/. For a random $p\times p$ symmetric matrix $M$, let $m_{\mu_M}(z)$ denote the Stieltjes ...
- 71
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How to interpret Stieltjes transform
I am new to Random Matrix Theory (RMT), and I am self reading the book Zhenyu Liao's book on RMT https://zhenyu-liao.github.io/book/.
A fundamental object of study is the so-called Stieltjes transform ...
- 71
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94
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Concentration inequalities for extreme singular values of complex Gaussian matrices
It is a classical result in non-asymptotic theory of random matrices that (See for example Rudelson and Vershynin (2010) https://arxiv.org/pdf/1003.2990#page=7#)
Let $A$ be an $N \times n$ ($N> n$)...
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132
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Minimum singular value of Gaussian random matrix/ inradius of Gaussian polytope
Let $X \in \mathbb{R}^{n \times N}$ be a random matrix with columns $X_1, \dots, X_N \sim N(0, I_n)$, independently.
Define the minimum $\ell_2^n \to \ell_\infty^N$ singular value
$$
s_{N, n} = \inf_{...
- 562
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1
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210
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Concentration inequality for quadratic form involving random matrix
I have $N$ i.i.d random vectors $\{X_k\}_{k=1}^N$ in $\mathbb{R}^n$ where each entry is bounded and positive.
I construct a matrix $M_N$ as
\begin{align}
M_N=\frac{1}{N}\sum_{k=1}^NX_kX_k^T
\end{align}...
- 103
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271
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Eigenvalue decomposition of empirical covariance matrix when $n<d$
Let $C \in \mathbb{R}^{d \times d}$ be a positive definite covariance matrix, and let $X_1, \ldots, X_n \sim N(0, C)$ be i.i.d. samples with $n < d$. Define the empirical covariance matrix:
$$\hat{...
- 109
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80
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Random matrices without limiting spectral density?
Consider the following sequence of random matrices.
Let $D_N = N\ diag(d_1, d_2, ..., d_N) / \sum_{n=1}^N d_n$ where $d_n=1/n$. So $Tr(D_N)=N$, but $D_N$ has eigenvalues that follow a $n^{-1}$ power ...
- 467
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Minimal value of Gaussian quadratic form
Let $X \in \mathbb{R}^{n \times N}$ be a Gaussian random matrix with independent standard Normal entries; assume $N > n$. Fix a unit vector $u \in \mathbb{R}^{n}$. For a subset $S$ of the integers ...
- 562
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Local laws for spiked matrices
There are Isotropic/Anisotropic local laws for Wigner matrices. I have also seen many works such as The Isotropic Semicircle Law and Deformation of
Wigner Matrices on analysis of eigenvalues of low ...
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117
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Change of variables of Laplacian on Hermitian matrices
Given the change of variables from the independent entries of an $N\times N$ Hermitian matrix $X$ to the coordinates of the eigenvalues and eigenvectors
$$\label{1}\tag{1}
X=U\Lambda U^{-1}
$$
The ...
- 31
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Minimal assumptions on distribution for computing expected Kronecker product
Let $ \mathbf{X} \sim D $ be a random matrix in $ \mathbb{R}^{n \times d} $, where $ D $ is some distribution. Assume that
$$
\mathbb{E} \left[ \mathbf{X}^{\top} \mathbf{X} \right] = \mathbf{\Sigma} \...
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204
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Matrix shuffling by shifting rows and columns
Let $M$ be an $n \times m$ matrix, with $1, \dots, m$ in the first row, $m+1, \dots, 2m$ in the second row, etc.
$$M = \left[
\begin{array}{c}
1 & 2 & \dots & m \\
m+1 & m+2 & \...
- 51
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111
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Function in Erdős–Yau section 17 (Edge Universality)
I'm currently reading about edge universality for random matrices in Erdős and Yau's book and in (17.6) of Corollary 17.3, there is the bound $\mathbb{P}[N(E,\infty)=0]\le\mathbb{E}[F(tr\chi_{E+\ell}*\...
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Slight skew in the distribution of eigenvalues for normal symmetric matrices
I've just started tinkering with random matrix theory, and to do so I've been performing some simulations in R.
Something I've noticed is that the distribution of eigenvalues shows slight skew from ...
- 31
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81
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Unbiased estimator of the singular values of a large matrix
Suppose I have a large matrix ${\bf X} \in {\Bbb R}^{m \times n}$. By independently sampling $r$ rows and $c$ columns of $\bf X$, we obtain a random submatrix $\bf S$. From $\bf S$, how to obtain an ...
- 451
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135
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Non-concentration implies full rank of random matrix over large field?
Consider a random $2 \times n$ matrix $X$ whose entries $X_{ij}$ take values in a finite field $E$ of characteristic $p$. Although the entries may not be independent, they satisfy the following non-...
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The Banach space structure of the span of the identity and the generators in the von Neumann algebra of free groups under 1-norm
I'll state the problem first:
Let $M = L(\mathbb{F}_n)$ be the group von Neumann algebra of the free group on $n$ generators. Let the $n$ free generators be $g_1, \cdots, g_n$. Is there a nice ...
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Example of an (infinite) matrix with a residual spectrum?
I am looking for an example of an (infinite) matrix that has a residual spectrum. For context, I know that the diagonal matrix $A$ with $A_{ii} = \frac{1}{i}$ has a point spectrum consisting of $\frac{...
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Localization of eigenvalues on complex plane
Let $B$ be a cyclic upper-triangular nonnegative matrix,
$$B = \begin{bmatrix} 0 & b_1 & 0& \dots &\dots &0 \\
0 & 0 & b_2 & 0& \dots & 0\\
\vdots &\vdots&...
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1
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168
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Variance of largest eigenvalue of Bernoulli matrix
We say that an $n\times n$ matrix $A_n$ is a random symmetric Bernoulli matrix if each entry $a_{ij}$ is $\pm 1$ with probability $1/2$, the entries $a_{ij}$ with $i\ge j$ are independent and $a_{ij}=...
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What is the implicit pseudorandomness conjecture behind the use of e.g. numpy.random() for probabilistic applications?
Suppose I want to investigate some complicated probabilistic phenomenon numerically, e.g. the eigenvalues of random matrices. One thing I might do is (ask some software to) generate a bunch of random ...
- 124k
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41
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In supercritical regime of BBP transition, the edge of bulk still follows Tracy-Widom?
In supercritical regime of BBP transition, where the largest eigenvalue is separated from the bulk, does the edge of bulk still follow Tracy-Widom? could another show me some reference? Thanks in ...
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133
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How to prove the solution of system of random equations concentrates on the solution of noiseless case?
Consider a set of unit vectors $\left\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_n \right\} \subset {\Bbb R}^2$ (that are not all in parallel). Let $\bf A$ be the adjacency matrix of a complete graph with ...
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73
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Trace exponential of sum of independent gaussians plus non commuting matrix
Let $Z_1$ and $Z_2$ be two independent standard normal random variables and $A_1$,$A_2$ two commuting matrices. Suppose that $B$ does not commute with either of them.
What tools does one have if one ...
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eigenvalues spectrum of random matrix expressions using determinant identities
Given a random matrix $X$ (e.g., with i.i.d. Gaussian entries) and two matrix expressions $A(X)$ and $B_\lambda(X)=B(X,\lambda)$ which satisfy (for any instance of X):
$$0=\det(\lambda I-A(X)) \iff 0=\...
- 333
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103
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Edge universality for Wigner matrix
I am trying to find a result stating that the rescaled eigenvalue point process at the spectral edge of a Wigner matrix converges to the determinantal Airy kernel point process.
I have found https://...
- 41
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47
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Dominance relationship and concentration of measure for random dual norms
I have two quadratic forms one of which is random and the other one is deterministic - I wonder if one dominates the other with 'high probability' which tends to 1 when dimension goes to infinity.
The ...
- 659
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1
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174
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Approximate Eigenvalue Distribution for Sum of Wishart Matrices
Let $\bar{\mathbf{W}}=\sum_{i=1}^{K} w_i \mathbf{W}_i$, where $\mathbf{W}_i \sim \mathcal{CW}_{M}(1,\mathbf{I}_M)$ is a complex Wishart matrix with $1$ degree of freedom and $\sum_{i=1}^{K} w_i = 1$ ...
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How to justify $\mathbf{S}_{x}= \frac{E_{s}}{N}I_{N\times N}$ as the best choice when the channel matrix is unknown?
I am a graduate student working in Wireless Communication, studying random matrix theory and its applications. In the context of determining channel capacity, I encountered the following generalized ...
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Efficient numerical schemes for Euler equations with negative pressure, or a complex version of Burger's equation
To be very short (before explaining more), I am trying to build an efficient and stable numerical scheme for the following systems of coupled PDEs:
$$\partial_t \rho + \partial_x[\rho v] = 0,$$
$$\...
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1
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Exercise on convergence spectral distribution of a random matrix
I am reading a book "Random Circulant Matrices" by Bose, Aurup and Saha, Koushik
and I am blocked by a simple statement which authors omit to prove.
Definitions. Let $A_n$ be a matrix (can ...
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Trace inequality involving random projections
Empirically I'm observing the following to hold for $s\ge 1$
$$\operatorname{Tr}(A^{2s})\le\|A^s\|_F^2\le\operatorname{Tr}(A^{s})$$
Where $A$ is a product of random projections
$$A=\prod_i^d I-v_iv_i^...
- 2,384
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70
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Matrix norm of a deterministic diagonal operator with sparse random gaussian non-diagonal entries
I am considering the usual matrix norm i.e. $||A||=\lambda_{max}(A)$ since our matrices are all symmetric.
We let $N$ be an integer and let $X_1,\dots X_{K_N}$ be a set of i.i.d centered gaussian ...
- 1
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0
answers
160
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Spectral distributions in Gram matrices built from unit random vectors in high dimensions (and relation to LLM embeddings)
I am wondering if any rigorous mathematical results are known for the following:
${V}_i \in \mathbb{R}^d$ ($i=1,2,\ldots N$) are randomly-distributed unit vectors, $\| V_i\| =1$.
Consider the ...
- 679
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Examples of random matrices that are not iid
There is a lot of literature on random matrices, however, in most of the sources that I have seen, the standard construction is by iid sampling of elements of the matrix. While it is natural from ...
- 659
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112
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Mean for product of random submatrix and its regularized inverse
I consider the following random circulant matrix
$$
H_w = F^* \mathrm{diag}(w_1, \dots w_p)F, \, w_i\overset{iid}{\sim}\Gamma(1,1),
$$
where $F$ is the matrix for discrete Fourier transform, $F^*$ ...
- 659
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207
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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}),
\...
- 311
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votes
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94
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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
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 562
2
votes
0
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213
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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 $C > 0$. ...
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488
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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 $\...
- 355
1
vote
0
answers
77
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 $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{...
- 11
1
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0
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98
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Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure
$$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$
for ...
- 131
3
votes
1
answer
179
views
Error bound for MonteCarlo estimate of elements in Gram-Matrix
Suppose I have a $n\times n$-symmetric positive-definite matrix $A$ with elements:
\begin{align}
[A]_{ij}=\int_{\Omega}f_i(x)f_j(x) \, dx, \quad i,j=1,\ldots,n
\end{align}
where $\Omega\subset \mathbb{...
- 103
2
votes
1
answer
106
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Parameters of Wishart distribution and generalized inverse
I recently came across the Wishart Distribution and a few things are unclear to me.
The Wikipedia page for the Wishart Distribution says that if $G=[g_1 \vert \; g_2\vert \; \ldots \vert g_n]$ is a $...
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votes
2
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412
views
Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 160
7
votes
1
answer
348
views
Smallest eigenvalue of a random matrix
Let $A \in \mathbb R^{n\times n}$ be a positive semi-definite matrix,
and let $b \in \mathbb R^n$. For a random vector $x \sim \mathcal N(0, I_{n\times n})$, consider the random matrices
$$
B_1 = A + ...
- 211
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votes
0
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87
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.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
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votes
0
answers
81
views
Why has the random Koopman matrix $ G_{xx}^{(-)} G_{yx} $ only eigenvalues on the complex unit circle?
Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution,
e.g.
$$ 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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1
vote
0
answers
116
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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 ...
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