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Decay of orthogonal contributions in a random set of vectors

Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$: $$\frac{v_1}{\|v_1\|},\...
Yaroslav Bulatov's user avatar
4 votes
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990 views

Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries

Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties: $M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative). The diagonal entries of $M$ ...
getraparth's user avatar
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47 views

"Probability" for a partitioned matrix to be singular

Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix $$ M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
Ludwig's user avatar
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132 views

Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix

Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
nikhil_vyas's user avatar