6

Only the subspaces mentioned by the OP are obviously closed. Let $V$ be a real vector space and $W\subset V$ a proper infinite-dimensional linear subspace. We shall endow $V$ with a norm so that $W$ will not be closed in $(V, \|\;.\;\|)$. For this we consider a Hamel basis for $W$ that we partition into a countable part $\{b_1, b_2, \dots\}$ and some (...


5

Linear maps on $\mathbb{R}^n \to \mathbb{R}^n$ are continuous and so, if one is invertible, also its sufficiently small perturbations are. It follows that invertible matrices form an open set in the space $\operatorname{M}_n$, hence non-invertible matrices form a closed set $C$. But $\operatorname{M}_n$ is a metrizable space, hence a perfectly normal space. ...


4

The state of the art appears to be reflected in Thompson, R. C. "On the eigenvalues of a product of unitary matrices I." Linear and Multilinear Algebra 2, 13 (1974) Chau, H. F. and Lam, Y. T. "Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices." Journal of Inequalities and ...


4

Here is an attempt to the problem for a worst-case $O$, with worse constants. So fix $O$, letting $o_i$ denote its $i$th row, and take $X$ random in $\{0,1\}^d$. We claim that $E |\langle o_i, X\rangle| \ge cst$. To see this, write $$\langle o_i, X\rangle = \langle o_i, \frac{{\bf 1}}{2}\rangle + \langle o_i, (X - \frac{{\bf 1}}{2})\rangle$$ and assume ...


3

The answer is negative, and this happens as soon as $n=2$. The question is whether the composition $X\mapsto L:=L_{X^2}$ is globally Lipschitz over ${\bf SPD}_n$. Let $x_j\in{\mathbb R}^n$ denote the $j$th column of $X$. Then we have the following formulae $$\ell_{11}=\|x_1\|,\quad \ell_{j1}=\frac{\langle x_1,x_j\rangle}{\|x_1\|}\,,$$ and so on, the ...


3

A productive way to approach this problem has been to focus on the real eigenvalues of the symmetric tensor and identify these with the critical points on the unit sphere of a Kostlan polynomial. In this way Paul Breiding was able to find an exact answer for the expected number of real eigenvalues when the symmetric tensor has a Gaussian distribution. The ...


3

This is a comment but I am not entitled. I don’t know if this the sort of thing you are looking for but I would suggest those subspaces which are closed in the weak topology $\sigma(V,W)$ where $W$ is the algebraic dual of $V$.


3

We prove the weaker bound $$ \mathbf{P} \left[ \|O \mathbb{x}\|_1 \leq \frac{cd}{\sqrt{\log d}} \right] \leq 2^{-Cd} $$ for some constants $C, c$. Define the Gaussian mean width of a compact subset $A \subset \mathbf{R}^d$ as $$ w(A) = \mathbf{E} \sup_{x \in A} \langle G,x \rangle $$ where $G$ is a standard Gaussian vector in $\mathbf{R}^d$. We use the ...


3

A more general problem is addressed in section 5 of Golub, G. H. "Some modified matrix eigenvalue problems". SIAM Review 15, 318 (1973). For background and other references see the Wikipedia article on the Bunch-Nielsen-Sorensen formula.


2

No. Example $$A=\left(\begin{array}{cc}-2&1\\ 1&-2\end{array}\right),\quad c=1.$$


1

Adding more detail to Mikael's point, the result seems to hold on average over $O$ because of the following: Using a Chernoff bound, we can see that the probability that for any constant $\epsilon$, with probability at least $1- e^{-C d}$ the random vector $x$ has at least $\frac{(1-\epsilon)d}{2}$ 1's. Consider a fixed vector $x \in \{0,1\}^d$. For a ...


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