# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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### How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]

Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$. I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$. My belief is that this is true is motivated by empirical ...
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### Lower bound for expectation of minimum eigenvalue

Let $X$ be a random (symmetric) matrix drawn from an unknown distribution. I have an estimate of $\lambda_{\min}(\mathbf{E}[X])$. Specifically, I have $$\lambda_{\min}(\mathbf{E}[X]) \geq c$$ a ...
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### Expectation of the inverse of random principal submatrices

The goal of this question is finding the concentration point of the inverse of random principal submatrices, which is posed as follows. Consider $\mathbf{S}\in\mathbb{S}^{n}_{++}$ to be a strictly ...
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### Johnson-Lindenstrauss with Orthogonalization

I have been looking at constructions satisfying the Johnson-Lindenstrauss Lemma (e.g., projections onto random subspaces, random Gaussian matrices, random Rademacher matrices, etc.). It seems that ...
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### What is the distribution of determinant of multi multiplication of some Gaussian matrices?

I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
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### The distribution of eigenvalues of linear combinations of random unitary matrices

Suppose that $\alpha_{1},\dots,\alpha_{r}$ are non-zero complex numbers. Let $U_{1},\dots,U_{r}$ be random $n\times n$-unitary matrices. Let $A=\alpha_{1}U_{1}+\dots+\alpha_{r}U_{r}$. I have observed ...
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### Do random asymmetric games have more complicated strategies than random symmetric games?

Let $\Delta \subset \mathbb R^n$ be the locus of vectors whose entries are nonnegative and sum to $1$. For $M$ an $n\times n$ matrix over $\mathbb R$, let $x_M \in \Delta$ be the vector $x$ that ...
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### Interpretation of Bai-Yin theorem and a question about (Hastie, Montanari, Rosset & Tibshirani)

Let $X_n\in \mathbb{R}^{p\times n}$ be a random matrix whose entries are i.i.d. $\mathcal{N}(0,1)$. Define $S_n = \frac{1}{n}X_n X_n^\top$. If $p/n\to y\in (0,1)$, the well-known Bai-Yin theorem ...
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