Questions tagged [matrices]

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43
votes
7answers
16k views

Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...
41
votes
3answers
5k views

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 ...
26
votes
1answer
934 views

Complexity of matrix powering

Let $M$ be a square integer matrix, and let $n$ be a positive integer. I am interested in the complexity of the following decision problem: Is the top-right entry of $M^n$ positive? Note that the ...
26
votes
0answers
559 views

Rank mod 6 vs rank over the reals

Let $A$ be a boolean matrix (eg with $0,1$ entries). Assume that $A$ has rank $\le r$ both over $\mathbb{F}_2$ and over $\mathbb{F}_3$. Does this imply that $A$ has low rank over the reals? This seems ...
25
votes
2answers
993 views

Approximating the sign rank of a matrix

The sign rank of a matrix A with +1,-1 entries is the least rank (over the reals) of a matrix B which has the same sign pattern as A (i.e., $A_{ij}B_{ij}>0$ for all $i,j$). This notion is important in ...
24
votes
1answer
3k views

Space complexity of Coppersmith–Winograd algorithm

Coppersmith–Winograd algorithm is the asymptotically fastest known algorithm for multiplying two $n \times n$ square matrices. The running time of their algorithm is $O(n^{2.376})$ which is the best ...
23
votes
2answers
685 views

Question about two matrices: Hadamard v. “the magical one” in the proof of the sensitivity conjecture

The recent and incredibly slick proof of the sensitivity conjecture relies on the explicit* construction of a matrix $A_n\in\{-1,0,1\}^{2^n\times 2^n}$, defined recursively as follows: $$A_1 = \begin{...
21
votes
0answers
682 views

Partial circulant matrices: Is there a non-zero vector $v\in \{-1,0,1\}^n$ such that $Mv=0$?

The following question arose as a side product of some work I have been part of recently. An $m$ by $n$ $(0,1)$-matrix $M$ is partial circulant if it can be formed by taking the first $m$ rows of a ...
20
votes
4answers
1k views

Positive topological ordering, take 3

Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix? This question is motivated by this problem: Positive topological ordering ...
20
votes
2answers
521 views

Explicit balanced matrix

Is it possible to build an explicit $N \times N$ $0/1$-matrix with $N^{1.5}$ ones such that every $N^{0.499} \times N^{0.499}$ submatrix contains less than $N^{0.501}$ ones? Or probably it is ...
19
votes
2answers
548 views

Bigger picture behind the choice of matrices in the Strassen algorithm

In the Strassen algorithm, to compute the product of two matrices $\mathbf{A}$ and $\mathbf{B}$, the matrices $\mathbf{A}$ and $\mathbf{B}$ are divided into $2 \times 2$ block matrices and the ...
19
votes
3answers
714 views

Complexity of deciding whether a matrix is totally regular

A matrix is called totally regular if all its square submatrices have full rank. Such matrices were used to construct superconcentrators. What is the complexity of deciding whether a given matrix is ...
19
votes
2answers
1k views

Can we decide whether a permanent has a unique term?

Suppose we are given an n by n matrix, M, with integer entries. Can we decide in P whether there is a permutation $\sigma$ such that for all permutations $\pi\ne\sigma$ we have $\Pi M_{i\sigma(i)}\ne \...
18
votes
2answers
715 views

Lower bounds on Gaussian complexity

Define the Gaussian complexity of an $n \times n$ matrix to be the minimal number of elementary row and column operations required to bring the matrix into upper-triangular form. This is a quantity ...
18
votes
0answers
410 views

In an $m$ by $n$ Boolean matrix, can you find a square block whose four corners are ones in $O(m \cdot n)$ time?

Decision Problem Input: An $m$ by $n$ Boolean matrix $M$. Decision Question: Does there exist a square block within $M$ such that upper-left corner entry == upper-right corner entry == lower-left ...
16
votes
1answer
2k views

How to compute powers of square matrices?

Suppose we are given a matrix $A \in \mathbb R^{N\times N}$, and let $m \in \mathbb N_0$. How fast can we compute the power $A^m$ of that matrix? The next best thing in comparison to computing $m$-...
16
votes
2answers
353 views

similar matrices

Given two $n \times n$ matrices $A$ and $B$, the problem of deciding if there exist a permutation matrix $P$ such that $B = P^{-1}AP$ is equivalent to GI(Graph ...
15
votes
2answers
5k views

What is the fastest algorithm to compute rank of a rectangular matrix?

Given an $m \times n$ matrix (assuming $m \ge n$), what is the fastest algorithm to compute its rank and basis of the columns? I am aware it can be solved through linear matroid intersection, which ...
14
votes
0answers
616 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, ...
13
votes
4answers
1k views

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 ...
13
votes
1answer
530 views

What's the complexity to check whether a matrix is Diagonalizable?

Given an $n\times n$ matrix $A$ with rational entries. What's the complexity to check $A$ is diagonalizable? I suspect that this can be done in P, but I do not know any reference. However, a more ...
13
votes
1answer
327 views

Cases of nearly linear time solvable linear systems

Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$ Solving for ${\bf x}$ through standard Gaussian Elimination ...
12
votes
2answers
810 views

Complexity of Membership-Testing for finite abelian groups

Consider the following abelian-subgroup membership-testing problem. Inputs: A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with arbitrary-...
11
votes
2answers
3k views

Complexity of Finding the Eigendecomposition of a *Symmetric* Matrix

This is a specialized version of a previous question: Complexity of Finding the Eigendecomposition of a Matrix . For NxN symmetric matrices, it is known that O(N^3) time suffices to compute the ...
11
votes
1answer
417 views

Constructing vectors in general position

Let a real $k\times n$ ($k\le n$) matrix ${\bf A}$ with the property that any collection of $k$ columns is full rank. Q: Is there an efficient way to deterministically find a vector ${\bf a}$ such ...
11
votes
0answers
199 views

the largest element of a matrix product

Given two matrices, I'm interested in finding the largest element of their product. I wonder if it's possible to do it significantly faster than the matrix multiplication the solution seems to require?...
11
votes
0answers
1k views

Complexity of finding the leading eigenvector of a graph Laplacian

Let ${\bf L}$ be the $n\times n$ Laplacian of a graph. What is the worst case complexity for calculating the maximum eigeinvector of ${\bf L}$? Are there any families of Laplacians for which it takes ...
10
votes
1answer
568 views

Can such a matrix exist?

During my work i came up with the following problem: I am trying to find an $n \times n$ $(0,1)$-matrix $M$, for any $n > 3$, with the following properties: The determinant of $M$ is even. For ...
10
votes
2answers
269 views

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 ...
10
votes
1answer
405 views

What is the asymptotically fastest known algorithm for computing the nullspace of a matrix?

I know Gaussian Elimination takes $O(n^3)$ arithmetic operations, but I'm unsure if any better algorithms are known.
10
votes
3answers
473 views

Linear programming solution in one pass with ordered variables

I have a family of linear programming problems: maximise $c' x$ subject to $A x\le b$, $x\ge0$. The elements of $A$, $b$, and $c$ are nonnegative integers, $c$ strictly positive. ($x$ should also ...
10
votes
2answers
433 views

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 ...
10
votes
1answer
360 views

What is the largest gap between rank and approximate rank?

We know that the log of the rank of a 0-1 matrix is the lower bound of deterministic communication complexity, and the log of the approximate rank is the lower bound of randomized communication ...
9
votes
1answer
725 views

Can we get a sorted list from a sorted matrix in $O(n^2)$

I'm confused. I want to prove that that the problem of sorting a $n$ by $n$ matrix i.e. the rows and columns are in ascending order is $\Omega(n^2\log n)$. I proceed by assuming that it can be done ...
9
votes
1answer
732 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? ...
9
votes
0answers
548 views

Finding SVD efficiently for $AB^T$

I have a low rank matrix given as $AB^T$ where $A,B \in \mathbb{R}^{n \times p}$ and $p \ll n$. (I know $A$ and $B$ separately) EDIT: (I have added the second question here since it was closed as a ...
8
votes
1answer
190 views

The complexity of the permanent of low rank matrices

I know that for an arbitrary $n \times n$ matrix, Ryser's algorithm can compute the permanent in $\mathcal{O}(2^n n^2)$ time. I'm interested in computing the permanent of $n \times n$ matrices of rank ...
8
votes
1answer
291 views

Transitive closure of an affine relation

I am looking for work on computing the transitive closure of an affine relation in the following sense: Let $R(x_1,\dots,x_n,x'_1,\dots,x'_n)$ be the relation defined by a system of linear ...
8
votes
1answer
5k views

What is computational complexity of calculating the Variance-Covariance Matrix?

I am using a calculation of the Variance-Covariance matrix in a program I wrote (for Principal Component Analysis), and am wondering what the complexity of it is. While obviously the Eigenvector ...
8
votes
0answers
189 views

Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
7
votes
2answers
365 views

Complexity of a variant of the max word problem. NP-complete?

I'd like to be able to state that the following problem is NP hard. I am wondering whether anybody have any pointers to related/recent work? The problem: Given a finite set of transition matrices $A$ ...
7
votes
1answer
353 views

Complexity to calculate a full set of eigenvectors over a finite field

Let a full rank $n\times n$ matrix ${\bf A}$ with elements over $\mathbb{GF}(2)$. What is the worst case complexity to calculate $n$ linearly independent (over $\mathbb{GF}(2)$) vectors, such that ...
7
votes
1answer
885 views

The computational complexity of spectral norm of a matrix

How hard is computing the spectral norm of a matrix? This paper says, ... it suffices to say that, except for few particular cases, the Matrix Norm problem is NP-hard. I expected that the ...
7
votes
1answer
518 views

Discrete log in GL(2,p)

Let $p$ be a large prime. Let $A$ be a $2\times 2$ matrix with coefficients in $GF(p)$ (i.e., coefficients taken modulo $p$). Let $B=A^k$, where $k$ is an integer not given to us. Given $p$, $A$, ...
7
votes
1answer
254 views

Using an oracle to find a vector $b$ for which $Ax=b$ has a solution

There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
7
votes
0answers
143 views

Reference for computing the rank of a matrix in polynomial time

In a recent paper, I need to use the fact that computing the rank of a matrix over the integers has polynomial complexity. Given the context, I don't particularly care about the exact asymptotics, as ...
7
votes
0answers
335 views

An algorithm to compute the number of paths of length at most k

So I had to answer the following question: Given a graph $G = (V, E)$, and two vertices $v_i, v_j$, compute the number of walks between $v_i$ and $v_j$ of length at most $k$. $G$ is not too large, ...
6
votes
1answer
801 views

Reducing the bandwidth of non-symmetric matrix

Is there an efficient algorithm to reduce the bandwidth of a directed graph's adjacency matrix? Something like the reverse Cuthill-McKee, but for non-symmetric matrices.
6
votes
1answer
132 views

What's the complexity of factoring over a set of generators (say in $GL_2$)?

In particular, if I have some char-0 field $k$ (let's take $\mathbb C$ for now) and I consider $G = GL_2(k)$ with arbitrary nontrivial distinct $A, B \in G$. Then for some $C \in GL_2(k)$ do we know ...
6
votes
1answer
236 views

Algebraic account of Gaussian elimination?

For fun, I've been looking at the interpretation of linear logic in terms of finite-dimensional vector spaces, and ran into an interesting question about the interpretation of double-negation-...