# Tag Info

### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 !...
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### 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 ...
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### 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

### 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/...
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### 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