# Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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### Dichotomy of the spectra of directed graphs

Compared to spectra of undirected graphs, which correspond to symmetric matrices, the spectra of directed graphs is not very well known: It is known that a directed graph $G = (V,E)$ has an adjacency ...
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### Complexity of reachability in linear dynamical systems over finite fields

Let $A$ be a matrix over the finite field $\mathbb{F}_2 = \{0,1\}$ and $x$, $y$ be vectors of the space $\mathbb{F}_2^n$. I am interested in the computational complexity of deciding whether there ...
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### Estimating the rank of a large sparse matrix

Consider a large sparse n by n matrix. Are there any methods to estimate its rank in time roughly proportional the number of elements in the matrix?
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### Quantum complexity of maximum inner product search

Given two matrices $X \in \mathbb{R}^{m \times k}$, $Y \in \mathbb{R}^{n \times k}$, maximum inner product search (MIPS) asks for the largest $l$ entries of $X Y^T$. Typically $k \ll m, n$ (many ...
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### Testing for satisfiability of a system of linear equations over GF(2)

Consider a system linear equations in $x$, $Ax =b$, where A is an $n\times n$ matrix, and $b$ is a column vector, and all operations are over $GF(2)$. Is it easier to check satisfiability of the ...
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### Min Hamming distance of a given string from substrings of another string

Let $U$ be a small finite set. Consider the following problem: Input: two strings $u \in U^k$ and $v\in U^n$ with $k \leq n$. Output: a (contiguous) substring of $v$ of length $k$ with the minimum ...
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### Inclusion of polytopes

Consider the following two system of linear (in)eqaulities: $S = Ax \leq b;\; Cx = e$ $T = Dx \leq d;\; Gx = g$ How can I check if $S\cap \neg T=\emptyset$ where $\neg T$ is the complement of the ...
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### Graph isomorphism problem with invertible adjacency matrices

This question is supplementary to the question asked here. One of the answers give a class of graphs for which the adjacency matrices are invertible which is defined as follows. Given a ...
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