# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

Statistics of spectral properties of matrix-valued random variables.

762
questions

0
votes

0
answers

8
views

Say that $W$ is a $m\times 1$ vector distributed $\mathcal{N}(0,\sigma^2 I)$.
Also, $X$ is a $n\times m$ Gaussian marix, $n<<m$, that is independent of $W$ with iid $\mathcal{N}(0,1)$ entries, ...

-1
votes

0
answers

98
views

What is the probability that the rank of the random matrix of size n x n generated by py is n? Please note that the probability that this rank is 0 is 1/(n+1)^n

0
votes

0
answers

35
views

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...

0
votes

0
answers

26
views

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with ...

3
votes

1
answer

563
views

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...

2
votes

2
answers

84
views

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).
Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...

0
votes

1
answer

76
views

Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda_1|>|\lambda_2|\ge \dots \ge |\lambda_n|$, can we get the order of $｜\lambda_1｜/｜\lambda_2｜$ or even $\lambda_1/\lambda_2$?
Any ...

0
votes

0
answers

22
views

I'm trying to follow the result in this work. Let me briefly introduce the problem. I have a $2N\times 2N$ matrix
$$
H=\begin{pmatrix}H_1 & V \\ V^{\dagger} & H_2\end{pmatrix}.
$$
Here both $...

0
votes

0
answers

44
views

I found that following equation holds for random vector $z \sim N(0,I)$ :
$\mathbb{E} [\frac{zz^T}{z^Tz}] = \frac{1}{n} I$
Proof is very simple that is only calculating integral for each component ...

2
votes

2
answers

200
views

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...

2
votes

1
answer

66
views

Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\...

0
votes

1
answer

39
views

In Chapter 3 of the textbook: An Introduction to Random Matrices, we have that for normalized GUE/GOE/GSE and ordering its eigenvalues $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$, we have that
$$
...

0
votes

0
answers

104
views

Let $r$ and $n$ be integers such that $1 \le r \ll n$, and $\|\cdot\|$ denote the Euclidean norm of vectors or the spectral norm of matrices.
Suppose that $\mathcal{D}$ is a probability distribution ...

2
votes

0
answers

18
views

Let $X$ be a standard Wishart matrix, i.e.,
$$
X = \sum_{j=1}^n g_j \otimes g_j \quad \mbox{where} \quad g_j \sim N(0, I_d).
$$
Above, $g_j$ are independent samples from the standard multivariate ...

5
votes

2
answers

298
views

Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties?
$F(P_\sigma) = \sigma$ for all $\sigma \in S_n$
$F^{...

1
vote

0
answers

58
views

I've seen a host of results concerning computations for $$\mathbb{E} \left[ \operatorname{tr} A^{i_1}\cdots \operatorname{tr} A^{i_j} \,\overline{\operatorname{tr} A^{k_1} \cdots \operatorname{tr} A^{...

3
votes

1
answer

114
views

For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.
...

1
vote

1
answer

163
views

Let $G$ be a symmetric Gaussian random matrix with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=\frac{1}{n}$, and ordering its eigenvalues $\lambda_1\le \lambda_2\le \dots \le \lambda_n$ corresponding ...

2
votes

1
answer

87
views

I've been playing around numerically with Haar random $\text{CUE}$ unitary matrices of size $N$ by $N$, with $N$ around $1000$. If I "truncate" the matrix by keeping the upper left $fN$ by $...

2
votes

1
answer

76
views

Given $H_1$ and $H_2$ i.i.d. $\mathit{GUE}$ matrices, what is the single eigenvalue distribution of $H_1 H_2 H_1$ in the large $N$ limit? This matrix is Hermitian, and so its eigenvalues are still ...

2
votes

1
answer

158
views

I have a matrix $A$ as follows:
$$
A=\begin{pmatrix}
0 & \boldsymbol{W} \\
\boldsymbol{W}^{\dagger} & \boldsymbol{H}
\end{pmatrix}
$$
where $H$ and $W$ are a random Hermitian $N\times N$ ...

1
vote

1
answer

84
views

Fix a Gaussian random matrix $A$ with $E[A_{ij}]=0$ for $i, j=1,\dots n$ and $E[A_{ij}^2]=\frac{1}{n}$. Let $v_1$ be the leading eigenvector of $A$. What is the non-asymptotic upper bound for $v_1$, ...

3
votes

1
answer

63
views

I am relatively new to the field of random matrices, and I suspect this may be relatively well-known.
Consider the real $N$ x $N$ matrix $O$ with i.i.d. standard normal entries, and consider the ...

1
vote

0
answers

58
views

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...

2
votes

1
answer

65
views

Suppose $x\in SG(\sigma^2)$ is a sub-Gaussian random vector, i.e.
$\left<u,x\right>\quad \forall u\in \mathbb{S}^{n-1}$ is a sub-Gaussian random variable.
My question is : what condition on the ...

0
votes

0
answers

10
views

Let $X$ be a random $d \times d$ Wigner matrix with entries of variance $1/d$ (that is, entries are i.i.d; for simplicity we can consider a GUE matrix). Is there a concentration bound on functions $f(\...

0
votes

0
answers

36
views

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function:
$$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...

1
vote

0
answers

73
views

Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...

0
votes

0
answers

29
views

Let $n$ and $d$ be large positive integers and let $\gamma \in (0,\infty)$.
Let $X$ be an $n \times d$ random matrix with iid entries from $N(0,1/d)$.
Let $A$ be an invertible deterministic $d \times ...

3
votes

0
answers

122
views

Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below?
$$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^...

2
votes

1
answer

65
views

I have a vector space which is a tensor product of two vector spaces, of dimensions $d_1, d_2$ respectively.
Consider Haar random unitaries acting on the full space with matrix elements $U_{i_1 j_1, ...

8
votes

1
answer

287
views

It is said by Halmos, P.R.; in "Lectures on ergodic theory"
"Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...

0
votes

1
answer

56
views

Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\...

0
votes

0
answers

29
views

One of the famous theorem in random matrix theory is one termed circular law. It is stated that if $a_{ij}$, for all $1\leq i\leq n, 1\leq j\leq n$,, is a famille of $i.i.d.$ complexe centered random ...

1
vote

0
answers

123
views

Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider
$$\sum_i x_i M_i$$
for random variables $x_i$.
What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$
...

0
votes

0
answers

31
views

The Tracy-Widom distribution gives the limiting distribution of the rescaled largest eigenvalue of a random matrix taken from an appropriate symmetry class. According to Bloemendal, the deformed Tracy-...

3
votes

0
answers

124
views

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...

0
votes

0
answers

79
views

Consider $x$ and $y$ two $N\times 1$ complex vectors, and $T$ a $N\times N$ complex random matrix. Each element of $T$ is chosen in a complex normal distribution. Let us also define the (Pearson) ...

0
votes

1
answer

148
views

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:
the distribution of spacings between Satake parameters of an L-function $F$ ...

1
vote

1
answer

158
views

Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of:
$$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...

8
votes

0
answers

207
views

Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...

0
votes

0
answers

28
views

Let $W \sim \mathcal{W}_d(V, n)$ be a random Wishart matrix. Let $A$ be a real symmetric positive definite matrix.
I am interested in computing
$$
\varphi_{V, d, n}(A) := \mathrm{tr}\big[\mathbb{E}[(W ...

1
vote

2
answers

97
views

Lets say $Y=\frac{1}{n}XX^\intercal$ and $X$ is a $n\times m$ random matrix whose entries are i.i.d gaussian. We know when $n$ and $m$ go to infinity with a fixed ratio, the singular values of $Y$ ...

1
vote

1
answer

63
views

Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...

3
votes

1
answer

300
views

While playing around with random matrices and I arrived at a different formula for the mean of the limiting normal distribution for a spectral CLT for sample covariance matrices. More precisely I have ...

3
votes

1
answer

110
views

The Tracy–Widom distributions admit many interpretations.
One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...

0
votes

1
answer

92
views

Let $d$, $n$, and $m$ be large positive integers. Let $X=(x_1,\ldots,x_n) \in \mathbb R^{n \times d}$ be a random matrix iid rows from some distribition $P$ on $\mathbb R^d$ which admits a density. ...

1
vote

1
answer

76
views

Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors.
I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value
$$\...

2
votes

1
answer

199
views

Let $n$ and $m$ be large positive integers. Let $x=(x_1,\ldots,x_n)$ be a vector of independent random variables from $N(0,1)$. It is clear that the covariance matrix of $x$ is $I_n$, the identity ...

1
vote

0
answers

43
views

Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices.
Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...