# Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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### Problems concerning subspaces of $M_{n}(\mathbb{Q})$

Let $M_{n}(\mathbb{Q})$ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q})$. Then if all the non-zero matrices in $N$ are invertible, what is ...
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### Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere

Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
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Consider irreducible nonnegative matrix $\mathbf{A} \in \mathbf{M}_{n}(\mathbb{R})$ such that $a_{ij} \in [0,1)$ as element of $\mathbf{A}$ of period $p$. If $\mathbf{A}^{T}$ is transpose of $\mathbf{... 1answer 97 views ### On the equation$[U, V] - V_x = C(x)$While considering the zero curvature equation$U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given$U(x)$and$C(x)$, find$V(x)$... 0answers 147 views ### Hurwitz–Radon problem for$ \mathbb{Q} ^{n} $What is the maximal number of orthogonal operators$ A _{1} , \dotsc, A _ {m} $in$ \mathbb{Q}^{ n } $satisfying the relations$ A_{i}^{2} = - I $and$ A_{i}A_{j} + A_{j}A_{i} = 0 $for$ i \neq ...
Given a matrix $P \in \mathbb{R}^{n \times d}$, we can get $P = U \Sigma V^T$ by using SVD. Let's say, we have another matrix $P' \in \mathbb{R}^{n \times d}$, it is the $P$ matrix with normalization ...