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If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$
Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
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1
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
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6
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Largest eigenvalues of a (random) correlation matrix?
I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
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