# Tagged Questions

For questions concerning random matrices.

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### If I generate a random matrix what is the probability of it to be singular?

Just a random question which came to my mind while watching a linear algebra lecture online. The lecturer said that MATLAB always generates non-singular matrices. I wish to know that in the space of ...
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### Finding “the” Marchenko-Pastur distribution in the original article of 1967

I am looking at distribution properties of eigenvalues of sample covariance matrices. Following the Wikipedia article on the Marchenko-Pastur distribution: Let $X$ denote a $M \times N$ random ...
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### Find a hypergeometric formula embracing three specific cases

For a parameter value $a=\frac{1}{4}$, I have the result Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}...
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### What is Steiltjes transform (other Integral transform) and how does it helps in probability theory, specifically in random matrix theory?

I have started growing interest in random matrix theory. Trying to understand it from "Random Matrices" by Madan Lal Mehta and "An Introduction to Random Matrices" by Anderson and many sources on ...
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### Intuitive explanation for Marcenko-Pastur law

I am looking for an intuitive reasoning behind the Marcenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
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### closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
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### Understanding the proof of Johnson-Lindenstrauss (JL) lemma

I have some questions about proof of Johnson-Lindenstrauss (JL) lemma. I appreciate any responses in advance. It is stated in the following paper: An elementary proof of JL lemma we argued: "Hence ...
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### Necessary/Sufficient conditions eigenvalues random matrix

I am working on the computational analysis of the eigenvalue spectra of real symmetric random matrices. At the moment my approach is fully empirical but I would like to develop a rough model of the ...
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### Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
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### Expected value of the smallest eigenvalue

Consider a random $m\times n$ matrix $M$ with elements from $\{-1,1\}$ and $m<n$. What is known about the expected value of the smallest eigenvalue of $MM^T$? The following picture shows ...
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### Numerical diagonalization of a random hermitian matrix $H=U\Lambda U^{-1}$: enforce uniqueness and uniformity of $U$

I've stumbled across this seemingly simple question, but I could not find a satisfactory answer. Suppose I have a complex hermitian random matrix $H$. It can be diagonalized by a unitary ...
### Eigenvalues of $AA^T$ for random circulant $A$
Consider a random circulant $n$ by $n$ 0-1 matrix $A$. Let $P(A_{1,i} = 1) = 1/\sqrt{n}$ and all the elements of the first row be independent. We know that the expected value of the diagonal ...