Questions tagged [random-matrices]

For questions concerning random matrices.

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems may be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

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Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?

I gave the following problem to students: Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$. Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same…
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probability for a $n\times n$ matrix to have no complex eigenvalues

Let $A$ be a $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has no complex eigenvalues? The answer cannot be 0 or 1, since the set of matrices with distinct real…
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How to generate random symmetric positive definite matrices using MATLAB?

Could anybody tell me how to generate random symmetric positive definite matrices using MATLAB?
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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 random matrices, what percentage are singular? Is…
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Expected rank of a random binary matrix?

Recently a friend stumbled across this question: Let $M$ be a random $n \times n$ matrix with entries in $\{0,1\}$ (both zero and one has probability $p = q = \frac{1}{2}$). What is its expected rank? My intuition is that it would be something of…
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Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a function of $n$? Does it tend to 1 as $n$ grows…
Joseph O'Rourke
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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
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What background is required to understand Random Matrix Theory

I would like to grasp RMT. I have knowledge on linear algebra, classical probability theory, Despite the above ones what should I study to understand RMT? Where to start? Is there any beginner's guide?
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A hole in the distribution of eigenvalues - is this real?

Plot the eigenvalues of many $n\times n$ real matrices in the complex plane, where the matrices are taken from a some distribution. For $n>7$, you can start to see a hole at the origin starting to form. From Girko's Circular Law I'd expect the…
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what is the variance of a constant matrix times a random vector?

$\newcommand{\Var}{\operatorname{Var}}$In this video is claimed that if the equation of errors in OLS is given by: $$u=y - X\beta$$ Then in the presence of heteroscedasticity the variance of $u$, will not be constant, $\sigma^2 \times I$, where $I$…
Mario GS
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Spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d. entries with variance $\frac{\sigma^2}{n}$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$? Here, $\rho(\cdot)$ denotes the largest…
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Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky that I have computed? Or will finding the Eigen…
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Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is equivalent of "taken a matrix at random from…
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Who first explicitly wrote the determinant identity $\det(1+AB) = \det(1+BA)$?

Though this identity can be easily proved, I am wondering who first explicitly write it in such a simple and elegant form? I check several textbooks on linear algebra but find no evidence (see below the list of books I have checked). The entry on…
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Maximizing trace of mixed products of two real symmetric matrices

Let $A$, $B$ be two $N \times N$ real symmetric matrices whose entries i.i.d.r.v. from a mean 0, variance 1 distribution. Let $I, J$ be even positive integers, and let $i_k, j_k$ for $k = 1,\ldots,n$ be arbitrary finite sequences of positive…
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