# Questions tagged [matrices]

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### Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
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216 views

### Checking properties of matrices

Given a sparse matrix $A$ in $\mathbb{Z}^{n\times n}$, how easily could one check whether a coefficient $\alpha_k$ of the characteristic polynomial $P_A$ of $A$ is equal to $0$ (without the need to ...
136 views

### How to efficiently generate a random 0-1 matrix of a given rank

How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
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### Matrix multiplication with transpose

Let $A,B\in\mathbb{F}^{n\times n}$ be two $n\times n$ matrices over the underlying field $\mathbb{F}$. In addition, $A$ is guaranteed to be a symmetric matrix, i.e, $A=A^{T}$. We assume complexity ...
133 views

### Low rank approximation of matrix under $l_2$ norm

Theorem 14 of this paper by Tam´as Sarl´os gives a relative error rank-$k$ approximation of a given matrix $A$ under the frobenius norm. I am looking for reference of a similar result (relative error ...
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### Finding the sparsest solution to a system of linear equations

How hard is it to find the sparsest solution to a system of linear equations? More formally, consider the following decision problem: Instance: A system of linear equations with integer coefficients ...
249 views

### Submatrix of small rank

Let $G=(V,E)$ be a graph with adjacency matrix $M=(m_{ij};i,j \in V )$ over $\mathbb{F}_2$ and $k \in \mathbb{Z^+}$. How can we find in polynomial time a subset $A \subseteq V$ such that The rank of ...
582 views

### Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
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### Matrix permanent is 0

Valiant's theorem says that computing the permanent of an $n\times n$ matrix is #P-hard. Is the problem of determining if a permanent is 0 any easier? This arises in the context of sequence A006063 in ...
181 views

### 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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### Matrix vector multiplication algorithm using minimal number of additions

Consider the following problem: Given a matrix $M$ we want to optimize the number of additions in the multiplication algorithm for computing $v \mapsto Mv$. I find this problem interesting ...
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### Arithmetic complexity of matrix powering

Assume $M\in\Bbb Z_{\geq0}[x_1,\dots,x_n]^{m\times m}$ be an $m\times m$ matrix in $n$ variables. We know that size of smallest formula that computes $\mathsf{Tr}(M^d)$ where $d\in\Bbb N$ could be ...
276 views

### “How much diagonal” a matrix is

I have a (biological) computational system that outputs squared matrices. These matrices will sometimes have a tendency to be diagonal-like, with higher values at and around the diagonal. I would ...
434 views

### Complexity of k-clique for hypergraphs

Classic Problem: Let a number $k$ be given. The $k$-clique problem is as follows. Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent? ...
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### Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
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### Products of PSD matrices that equal an orthogonal matrix

You have k symmetric real n-by-n matrices A_1...A_k, each bounded by some parameter eps < 1/2 in spectral norm. You know that the product (Id+A_1)(Id+A_2)...(Id+A_k) equals an orthogonal matrix Y. ...
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### Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
125 views

### Minimal set of Hyperrectangles covering an n-dimensional binary matrix with row permutations

My input is a n-dimensional binary matrix. My goal is to find the set of Hyperrectangles that covers every '1' at least once and covers not a single '0', which has minimal cardinality (the least ...
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### Matrix of ranking

In any graph of n nodes in any dimension, define matrix $M_r$ of ranking as $\forall r_{ij}\in M_r$, $r_{ij}$ is the ranking of j to i. That is, j is the $r_{ij}$th nearest node to i. Therefore, any ...
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### Find non-intersecting submatrices

I have a rectangular boolean matrix and I'd like to have an efficient algorithm to find non-intersecting submatrices. I'll to demonstrate that in the example below. The ideal case is when all ...
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### Rate of convergence for the Perron–Frobenius theorem

The Perron–Frobenius Theorem states the following. Let $A = (a_{ij})$ be an $n \times n$ irreducible, non-negative matrix ($a_{ij} \geq 0, \forall i,j: 1\leq i,j \leq n$). Then the following ...
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### Low rank Log rank conjecture

What is known about log rank conjecture in special situations of $O(\log N)$ rank $0/1$ matrix of size $N\times N$? Is there at least a conditional result showing better than $O(\sqrt{\log N})$ bound?
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### Parameterized Complexity of Minimum Type Selection

Consider the following problem that I call »Minimum Type Selection«: Input: $k$ sets of bit vectors, each of length $n$ and a number $l$. Question: Is it possible to pick exactly one bit vector from ...
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### Is there a diagonal matrix D such that DMD is SDD, where M is SPD matrix

Let $M$ be symmetric and positive definite matrix (SPD). It is known  that if $M$ is SPD and in addition satisfies $M_{ij}\leq 0$, for $i\neq j$ (called M-matrix) then there is a positive ...
46 views

### Reference for Nuclear Norm Relaxations

I have seen a bunch of results concerning Matrix Completion, PCA, Compressed Sensing where a common theme has been to relax the Rank constraint/objective by replacing it with Nuclear Norm. I was ...