Questions tagged [matrices]

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46 votes
8 answers
21k 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 ...
Lev Reyzin's user avatar
  • 12k
44 votes
3 answers
6k 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 ...
Brian's user avatar
  • 541
26 votes
1 answer
980 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 ...
Joel's user avatar
  • 497
26 votes
0 answers
703 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 ...
Shachar Lovett's user avatar
25 votes
2 answers
1k 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 ...
Moritz's user avatar
  • 1,714
24 votes
1 answer
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 ...
Shiva Kintali's user avatar
23 votes
2 answers
743 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{...
Clement C.'s user avatar
  • 4,451
22 votes
3 answers
754 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 ...
user avatar
20 votes
4 answers
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 ...
domotorp's user avatar
  • 13.8k
20 votes
2 answers
533 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 ...
ilyaraz's user avatar
  • 1,569
20 votes
0 answers
705 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 ...
Simd's user avatar
  • 3,902
19 votes
3 answers
789 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 ...
Markus Bläser's user avatar
19 votes
2 answers
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 \...
domotorp's user avatar
  • 13.8k
18 votes
2 answers
937 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 ...
Moritz's user avatar
  • 1,714
18 votes
0 answers
462 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 ...
Michael Wehar's user avatar
17 votes
2 answers
437 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 ...
DurgaDatta's user avatar
  • 1,281
16 votes
1 answer
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 $m$-...
shuhalo's user avatar
  • 1,165
16 votes
2 answers
7k 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 ...
Ho Yee Cheung's user avatar
14 votes
0 answers
685 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, ...
Michael Wehar's user avatar
13 votes
4 answers
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 ...
Michael Wehar's user avatar
13 votes
2 answers
891 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-...
13 votes
1 answer
583 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 ...
amatrix's user avatar
  • 133
13 votes
1 answer
332 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 ...
Dimitris's user avatar
  • 1,346
12 votes
0 answers
274 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?...
MWB's user avatar
  • 231
11 votes
3 answers
515 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 ...
Robert Almgren's user avatar
11 votes
2 answers
4k 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 ...
Lihong Li's user avatar
  • 111
11 votes
1 answer
474 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 ...
Dimitris's user avatar
  • 1,346
11 votes
0 answers
2k 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 ...
Dimitris's user avatar
  • 1,346
10 votes
1 answer
596 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 ...
Etsch's user avatar
  • 615
10 votes
2 answers
310 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 ...
vzn's user avatar
  • 11k
10 votes
1 answer
807 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 ...
user2038833's user avatar
10 votes
1 answer
944 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? ...
Michael Wehar's user avatar
10 votes
1 answer
426 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.
GMB's user avatar
  • 2,393
10 votes
1 answer
255 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 ...
Teferi's user avatar
  • 103
10 votes
2 answers
544 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 ...
john mangual's user avatar
10 votes
1 answer
407 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 ...
pyao's user avatar
  • 425
9 votes
0 answers
572 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 ...
user avatar
8 votes
1 answer
295 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 ...
warakawa's user avatar
  • 233
8 votes
1 answer
7k 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 ...
Yet Another Geek's user avatar
8 votes
0 answers
195 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 $w_{ij}$ ...
Rasto S.'s user avatar
  • 101
7 votes
2 answers
377 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 $A$ ...
Al-Mousa's user avatar
7 votes
1 answer
379 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 ...
Dimitris's user avatar
  • 1,346
7 votes
1 answer
1k 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 ...
Omar Shehab's user avatar
7 votes
1 answer
1k 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 $p$, $A$, ...
D.W.'s user avatar
  • 12k
7 votes
1 answer
270 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 ...
Uthsav Chitra's user avatar
7 votes
0 answers
268 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 ...
Jakub Konieczny's user avatar
7 votes
0 answers
1k views

Best algorithm for inversion of symmetric positive-definite matrices

As I know, the time complexity for usual inversion algorithm of general matrix is $O(n^3)$. In recent years, there is a improvement from $O(n^3)$ to $O(n^{2.8....})$. However, does there exists a ...
Timespace's user avatar
  • 171
7 votes
0 answers
506 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, ...
Federico Lebrón's user avatar
6 votes
1 answer
1k 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.
Jonathan H's user avatar
6 votes
1 answer
141 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 ...
Ben Kushigian's user avatar