Skip to main content

All Questions

Filter by
Sorted by
Tagged with
42 votes
3 answers
5k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
Denis Serre's user avatar
  • 52.3k
25 votes
1 answer
4k views

What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
Qiaochu Yuan's user avatar
3 votes
1 answer
3k views

Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
valle's user avatar
  • 884
19 votes
1 answer
2k views

Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following: Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
Suvrit's user avatar
  • 28.6k
18 votes
1 answer
1k views

How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
sbahmani's user avatar
  • 181
12 votes
1 answer
3k views

Matrix inversion lemma with pseudoinverses

The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own. Suppose we pick $n$ values $x_1,\...
Suvrit's user avatar
  • 28.6k
10 votes
2 answers
1k views

Probability of random (0,1) Toeplitz matrix being invertible

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. What is the probability that a random $n \times n$ binary Toeplitz ...
user avatar
8 votes
2 answers
2k views

Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
Atnap's user avatar
  • 127
8 votes
0 answers
232 views

Decay of orthogonal contributions in a random set of vectors

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\|},\...
Yaroslav Bulatov's user avatar
6 votes
1 answer
299 views

Phase transition in matrix

Playing around with Matlab I noticed something very peculiar: Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by $$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$ ...
Sascha's user avatar
  • 536
5 votes
0 answers
235 views

Riemann theta function inequality for a class of large random matrices

The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
Dierk Bormann's user avatar
3 votes
0 answers
225 views

Eigenvalues of Hadamard product of two Wishart-type matrices

Given two independent Gaussian matrices with i.i.d. entries: $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times q}$, where and $A_{i,j},B_{i,j}\sim\mathcal{N}(0,1)$. Assume that $\max(p,q)<n....
M-Brust's user avatar
  • 31
3 votes
0 answers
202 views

Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices

I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
Dierk Bormann's user avatar
3 votes
0 answers
130 views

The probability that the dominant eigenvalue of a random real matrix is real

Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
Joseph Van Name's user avatar
3 votes
1 answer
655 views

Upper bounds on the condition number of the eigenvector matrix

Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$. Question: Are there any upper bounds on the condition number of the ...
Ludwig's user avatar
  • 2,712
2 votes
0 answers
172 views

Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal

Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
Juan's user avatar
  • 61
2 votes
0 answers
96 views

Smallest eigenvalue separation in the Gaussian ensemble of random matrices

This question is motivated by Guido Li's question Expected minimal distance of eigenvalues, which concerns the sum of a deterministic matrix and a Gaussian random matrix. The deformation of the ...
Carlo Beenakker's user avatar
2 votes
1 answer
244 views

Expected minimal distance of eigenvalues

Let $A$ be an arbitrary symmetric matrix and $B$ be a random GUE matrix. I would like to know. Are there any results on the minimal eigenvalue distance between two distinct eigenvalues of $A+B$? I ...
Guido Li's user avatar
2 votes
1 answer
905 views

Diagonalizability of Gaussian random matrices

Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
user50394's user avatar
  • 123
1 vote
1 answer
84 views

Asymptotic property of the left singular vectors of i.i.d. data matrix

Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments: $$ \mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top. ...
Seung Hyeon Yu's user avatar
1 vote
1 answer
1k views

product of Gaussian random matrix and a deterministic diagonal matrix

Suppose that $G$ is an $n\times n$ Gaussian random matrix of i.i.d. entries $N(0,1/n)$ and $D$ is an $n\times n$ deterministic diagonal elements. I'd like to know if there have been results on the ...
user58955's user avatar
  • 640