32
votes

### For positive definite $A,B$ why does $AB+BA$ tend to be positive definite?

Your question appears to be based on a false premise. In fact $AB+BA$ does not tend to be positive definite as $n$ increases, even within the particular distribution you happen to be using.
To ...

- 1,272

31
votes

Accepted

### What is the Katz-Sarnak philosophy?

The "Katz-Sarnak philosophy" is just the idea that statistics of various kinds for $L$-functions should, in the large scale limit, match statistics for large random matrices from some particular ...

- 45k

22
votes

### Expected value of determinant of simple infinite random matrix

Very nice problem!
Let me recall you that the determinant of $n \times n$ matrices with entries in $\{0,1\}$ is related to the one of $n+1 \times n+1$ matrices with entries in $\{-1,+1\}$: replace ...

- 1,755

18
votes

### What is the Katz-Sarnak philosophy?

I'm going to give an answer that discusses some things that the other answers don't go into as much detail on. In particular let me try to explain why the results you mention on classical groups ...

- 122k

16
votes

Accepted

### For positive definite $A,B$ why does $AB+BA$ tend to be positive definite?

$\text{tr}(AB+BA) = 2 \operatorname{tr}(A^{1/2} B A^{1/2}) > 0$, so that may produce some bias toward positive eigenvalues. In particular if you generate your "random" matrices in such a ...

- 51.8k

16
votes

Accepted

### Counting eigenvalues without diagonalizing a matrix

Here is an efficient method.
First of all, I must quote that diagonalizing $M$ is not a method, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial !...

- 48.5k

16
votes

Accepted

### Relative Entropy and p-norm

The argument below is not very elegant,but it is, indeed, a standard exercise. Let $g=\max(f-1,0)$. We shall prove that
$$
f\log f\le 2g+\frac 2{p-1}g^p\,.
$$
The integration and Holder then give the ...

- 54.4k

15
votes

Accepted

### Expected size of determinant of $AA^T$ for non-square random $A$

By the Cauchy-Binet theorem, $\det AA^T=\sum (\det B)^2$, where $B$
ranges over all $m\times m$ submatrices of $A$. The expected value of
$(\det B)^2$ is $(m+1)!/4^m$, so the expected value of $\det ...

- 45.9k

12
votes

### Jensen Polynomials for the Riemann Zeta Function

The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, ...

- 122k

11
votes

### What kind of random matrices have rapidly decaying singular values?

I hope I understood the OP correctly, in case not please let me know. And I will discuss the case of eigenvalue instead of singular value without much loss of generality.
In case you are only ...

- 7,593

11
votes

Accepted

### Gaussian integrals over the space of symmetric matrices

A recursion formula for the moments of the Gaussian orthogonal ensemble, M. Ledoux (2009).
The desired recursion formula for the moment $b_p^N\equiv E\,[\,{\rm tr}\,(S_N^{2p})]$ is
I notice a ...

- 155k

11
votes

### What is the Katz-Sarnak philosophy?

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by ...

- 2,440

11
votes

Accepted

### Average of the maximum matrix element over the Haar measure

The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting:
$$\int dU \max_j |U_{1,j}|...

- 155k

11
votes

### Number of permutations with longest increasing subsequences of length at most $n$

There is an explicit determinental formula for these numbers due to Gessel in Symmetric functions and P-recursiveness (JCTA, 1990). Asymptotics were known much earlier and appear in a paper by Amitai ...

- 1,295

10
votes

Accepted

### Expectation of trace of nth power of unitary matrices

$$\int_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$
see Theorem 2.1.b of Diaconis and Evans (2001). [*]
[*] This 2001 reference corrects an earlier paper by Diaconis and Shahshahani (...

- 155k

10
votes

Accepted

### Riemann zeta function: pair correlations vs. neighbor spacings

This next-nearest-neighbor distribution of the Riemann zero's is addressed in Mehta's book on random-matrix theory. It is well reproduced by that of the Gaussian Unitary Ensemble (GUE), compare black ...

- 155k

10
votes

### Expected value of determinant of simple infinite random matrix

I agree with user39115!
I will give a heuristic from random matrix theory because we know the global behaviour of the eigenvalue. First
$$A=p 1 +\sqrt{N(p-p^2)}\frac{B}{\sqrt{N}} $$ where $1$ is the ...

- 4,266

10
votes

Accepted

### Scaling in Mehta's integral

Yes, this follows by the de la Vallée-Poussin necessary and sufficient condition for the uniform integrability. Indeed, suppose that
\begin{equation}
\gamma n^2\to a
\end{equation}
(as $n\to\infty$) ...

- 85.4k

10
votes

Accepted

### Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

Call $S$ the set of matrices with repeated eigenvalues and fix a hermitian matrix $A\not\in S$. In the vector space of hermitian matrices, any line through $A$ intersects $S$ in at most finitely many ...

- 3,056

9
votes

### The probability for a symmetric matrix to be positive definite

I came across this question while preparing a talk on this paper. There we gave the explicit formula for $p_n$ (in the GOE) that the other answers anticipated could be found by random matrix methods.
...

- 91

9
votes

### Moments of the trace of orthogonal matrices

This answer is a follow-up to the other answers (particularly to Carlo Beenakker's answer from September 4).
First off, Carlo's conjecture is indeed true. That is:
Theorem. If $n \geq k$ then
$$...

- 5,088

9
votes

### The expected square of the determinant of a random row stochastic matrix

Hidden in the comment by Victor Kleptsyn is a really nice argument. Since nobody upvoted that comment yet (I just did) it's probably worth expanding it.
From a probabilistic approach it's more ...

- 4,298

9
votes

Accepted

### A question about the paper "The Condition Number of a Randomly Perturbed Matrix"

One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough.
Thanks for pointing out this ...

- 93.8k

9
votes

### Jensen Polynomials for the Riemann Zeta Function

Unfortunately, I cannot comment.
https://www.youtube.com/watch?v=HAx_pKUUqug 56:28

- 89

9
votes

Accepted

### Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$

We have
$$ C(W) = 2 A \circ A + v v^\top$$
where $v$ is the vector with entries $\|w_i\|^2$, $A$ is the Wishart matrix with entries $w_i^\top w_j$, and $\circ$ is the Hadamard product. From the Schur ...

- 93.8k

9
votes

### Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

Here is a (I think) mathematically correct, but clearly morally wrong, answer via extreme overkill.
Upon multiplying by $i$, we may work instead with skew-Hermitian matrices, i.e., with $\mathfrak u(n)...

- 9,077

8
votes

Accepted

### Is there a way to simplify the following trace expression?

After a cyclic permutation of the trace, the expression you need is
$$Y=\text{tr}\left\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}_{M\times M} & \mathbf{0}_{M\times N} \\ \mathbf{0}_{N\...

- 155k

8
votes

### GOE/GSE duality and Bott periodicity

there is something called the Altland Zirnbauer classification of topological insulators which is related to both the random matrices and Bott periodicity
https://golem.ph.utexas.edu/category/2014/...

- 21.9k

8
votes

Accepted

### GOE/GSE duality and Bott periodicity

The entire set of correspondences can be read off from this table:
Listed are the 10 symmetric spaces and for each space in the left column the dual space is shown in the right column, as explained ...

- 155k

8
votes

Accepted

### References for reasoning about the spectrum of a convex body?

A very direct result giving characteristic function control for uniform measures on compact convex sets and hence spectrum is [Kulikova& Prokhorov].
It sounds to me that all you want to study ...

- 7,593

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