All Questions
14 questions
2
votes
0
answers
269
views
Singular values of Kronecker product of random matrices
I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
- 121
8
votes
1
answer
325
views
On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k
14
votes
1
answer
417
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
1
vote
1
answer
160
views
Estimates of product of eigenvalues gaps for Wigner matrices
Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$.
My question. For any fixed $i\in\{1,\dots,n\}$, ...
- 2,712
2
votes
2
answers
248
views
On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
5
votes
1
answer
274
views
Precise probability that $m$ random vectors in $n$ dimensional space are nearly-orthogonal
Consider $m$ vectors $v_1,\dots,v_m$ in $\mathbb R^n$, drawn uniformly and independetly from unit sphere. It is pretty straightforward from Chebyshev inequality that
$$
\mathrm P (\forall i\ne j \ |...
- 331
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 ...
- 884
3
votes
1
answer
777
views
Lower bound of the expectation of the product of inner products of random vectors
I encountered the following value in my research:
Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$.
Denote
$$
L = \mathop{\mathrm{E}}_x[ \prod_{1\...
- 1,551
2
votes
0
answers
102
views
Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries
I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
- 121
4
votes
2
answers
2k
views
Advanced reference and roadmap about random matrices theory
There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question.
I really want to hear ...
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 387
27
votes
3
answers
13k
views
What is known about the distribution of eigenvectors of random matrices?
Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...
- 433
4
votes
1
answer
214
views
The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)
During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...
- 41
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181