# Tagged Questions

Statistics of spectral properties of matrix-valued random variables.

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### Trace of the inverse sample covariance as the number of samples and dimension scale to infinity

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_{p\times p})$, with $n>p$. Let $\hat S=\frac1n\sum_{i=1}^n x_i x_i^T$ be the sample covariance. Assume the asymptotic setting where $\frac pn\to \alpha<1$. ...
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### Average minimum number of random k-sparse vectors in $\mathbb{F}_2^n$ to span a specific base vector?

A while back I posted a question in MO about the average minimum number of independent random k-sparse (having at most $k$ non-zero elements) vectors belonging to $\mathbb{F}_2^n$ to span the whole ...
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### Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$? EDIT: As said in the comments, I'm looking for the ...
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### Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
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### When is $\left\|\hat S\left(\hat S+T\right)^{-1}\right\|_2\le 1$ for p.d. $S$ and $T$?

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_{p\times p})$. Let $S$ be the covariance of the $x_i$, $\hat S=\frac1n\sum_{i=1}^n x_ix_i^T$. What is the set of positive-definite $p\times p$ matrices $T$ such ...
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### A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...
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143 views