Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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3
votes
0answers
177 views

An interesting construction of a Tits building?

The notion of Tits building was introduced by Jacques Tits to study certain questions in group theory. The wikipedia entry gives a way to construct a Tits building from a vector space, but I would be ...
4
votes
0answers
169 views

Extensions of Sylvester's inertia law?

Sylvester's inertia law deals with the signatures of quadratic forms. I was thinking that it may be possible to extend this to multilinear forms; here is a first attempt. Let $M$ be a $k$-linear form ...
3
votes
1answer
292 views

Parallel (NC) replacements for Gaussian elimination?

Suppose one has an matrix $A$ with $c$ columns and $r$ rows with entries in the binary field $GF(2)$. One wants to determine its rank, and a basis for its null-space. These can be computed easily ...
19
votes
3answers
645 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 ...
4
votes
0answers
384 views

Bounding the spectral radius of a sub-stochastic matrix

Suppose that I have a "sub-stochastic" matrix, namely, for an $n\times n$ matrix $A$ with nonnegative entries such that for any $i$, $\sum_j a_{ij}\leq 1$ and there exists some $i$ with $\sum_j a_{ij}&...
3
votes
1answer
107 views

compleixty of rational checking of eigenvalues

Given a matrix $A$ with rational entries, how to check whether all the eigenvalues of $A$ are rational? What's the complexity of this problem? It seems that this can be done in polynomial time, but ...
27
votes
3answers
1k views

Decide whether a matrix's kernel contains any non-zero vector all of whose entries are -1, 0, or 1

Given an $m$ by $n$ binary matrix $M$ (entries are $0$ or $1$), the problem is to determine if there exists two binary vectors $v_1 \ne v_2$ such that $Mv_1 = Mv_2$ (all operations performed over $\...
1
vote
0answers
128 views

Constructing a digraph from its spectrum

This is related to the following question which has already been explored: Reverse Graph Spectra Problem? So it seems as if given a sequence of real numbers, it is not always possible to generate a 0-...
10
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0answers
161 views

Number of graphs with prescribed spectrum

I have a question relevant to the number of graphs with prescribed spectral ratio. Let $A$ be the adjacency matrix of a graph on $n$ vertices. Let $\lambda_i$ be its $i$-th largest (signed) eigenvalue....
3
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0answers
85 views

Finding the closest subspace to a collection of subspaces

Suppose we have a collection of linear subspaces $\mathbb{C}$ lying in $\mathbb{R}^d$, such that each $c \in \mathbb{C}$ is of dimension at most $k \leq d$ for a given fixed $k$ and $|\mathbb{C}| = n$....
1
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0answers
443 views

Time complexity of clustering based on random walk

What is the time complexity of the following algorithm (from this paper suggested by Zhou) to partition directed graph? Can I use the complexity of eigen vector computation for this purpose? The ...
18
votes
3answers
2k views

Determinant modulo m

What are the known efficient algorithms for computing a determinant of an integer matrix with coefficients in $\mathbb{Z}_m$, the ring of residues modulo $m$. The number $m$ may not be prime but ...
13
votes
1answer
468 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 ...
1
vote
1answer
954 views

What is the advantage of a transformation matrix in perspective projection?

I was messing around with perspective projection. And it got me wondering whether using a perspective transformation matrix is the most efficient way. Because I also came up with another method which ...
17
votes
2answers
718 views

What is the space complexity of calculating Eigenvalues?

I am looking for a survey paper or a book covering results about the space complexity of common linear algebra operations such as matrix rank, eigenvalues calculation, etc. I stress the "space ...
2
votes
0answers
233 views

Running time of Jacobi vs Golub-Kahan for SVD

For an $m \times n$ matrix, what is the running time for computing the Singular Value Decomposition (SVD for short) via Jacobi's method, and Golub-Kahan? The source I read mentions that Jacobi's ...
18
votes
4answers
851 views

Checking if all products of a set of matrices eventually equal zero

I am interested in the following problem: given integer matrices $A_1,A_2, \ldots, A_k$ decide if every infinite product of these matrices eventually equals the zero matrix. This means exactly what ...
4
votes
1answer
320 views

state-of-the-art bit complexity of the determinant

I'm trying to understand the full bit-complexity of computing the determinant of an $n\times n$ integer matrix, with each entry represented by $M$ bits. I would like to know what is the state-of-the-...
-3
votes
1answer
264 views

the product of a matrix and a permutation matrix [closed]

Can a permutation matrix (P) be used to change the rank of another matrix (M)? Is there any literature to this effect, or to the contrary? I've tried a few small examples and the resulting matrix (M2)...
2
votes
0answers
242 views

normalized tensor rank

Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)? For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has ...
11
votes
0answers
193 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?...
0
votes
1answer
221 views

Various conjectures which is similar to Log Rank conjecture

Log rank conjecture is one of the most famous open problems in the area of communication compleixty. Lets consider the two party cdommunication complexity. Alice and Bob have $n$ bit strings $a,b$ , ...
7
votes
0answers
323 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, ...
19
votes
4answers
636 views

How to obtain the unknown values $a_i,b_j$ given an unordered list of $a_i-b_j\mod N$?

Can anyone help me with the following problem? I want to find some values $a_i,b_j$ (mod $N$) where $i=1,2,…,K, j=1,2,…,K $ (for example $K=6$), given a list of $K^2$ values that correspond to the ...
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 $w_{ij}$ ...
10
votes
1answer
484 views

Why does the log-rank conjecture use rank over the reals?

In communication complexity, the log-rank conjecture states that $$cc(M) = (\log rk(M))^{O(1)}$$ Where $cc(M)$ is the communication complexity of $M(x,y)$ and $rk(M)$ is the rank of $M$ (as a matrix)...
13
votes
0answers
182 views

Complexity to compute the eigenvalue signs of the adjacency matrix

Let $A$ be the $n\times n$ adjacency matrix of a (non-bipartite) graph. Assume that we are given the amplitudes of its eigenvalues, i.e., $|\lambda_1|=a_1,\ldots, |\lambda_n|=a_n$, and we would like ...
17
votes
2answers
435 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
1answer
764 views

Is solving systems of equations modulo $k$ in $\mathsf{coMod}_k\mathsf L$ for $k$ composite?

I'm interested in the complexity of solving linear equations modulo k, for arbitrary k (and with a special interest in prime powers), specifically: Problem. For a given system of $m$ linear ...
1
vote
0answers
238 views

application for the Kchinchine inequality in Computer Science

The classical Kchinchine inequality states that for vector $a=(a_1, \ldots, a_{2m})\in R^{2m}$, for $p\geq 2$, and for independent Rademacher random variables $r_1, \ldots, r_{2m}$, one has $$ E(|\...
11
votes
2answers
265 views

On $n$ dimensional manifolds and lattices

EDIT (By Tara B): I'd still be interested in a reference to a proof of this, as I had to prove it myself for my own paper. I'm looking for the proof of Theorem 4 that appears in this paper: An ...
3
votes
1answer
259 views

What is this matrix column-selection problem, and how hard is it to approximate?

I ran across the following simple-to-state problem involving selection of a subset of columns simultaneously for a number of matrices. I suspect it might be well known, though I can't seem to place it....
2
votes
1answer
363 views

All Pairs Shortest Path - Directed graph with integer weights

I don't understand how Distance Product works (or Min Plus Product). If we replace each argument in $A$ from $a_{i,j}$ to $x^{a_{i,j}}$ and each argument in $B$ from $b_{i,j}$ to $x^{b_{i,j}}$ and ...
2
votes
2answers
3k views

Minimizing the maximum dot product among k unit vectors in an n-dimensional space

Suppose, we are given a set of $k$ unit vectors $v_1,\ldots,v_k$ in $\mathbb{R}^n$. Consider all possible dot products among distinct vectors $v_i \cdot v_j$, where $i \ne j$. Let, $$\alpha = \max_{1 ...
1
vote
0answers
158 views

Trace minimization with an orthogonality constraint

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under the constraint: that $X$ is orthogonal. All the matrices have real entries and $A,B$ are ...
13
votes
1answer
476 views

Algorithmic Vector Problem

I have an algebraic problem related to vectors in the field GF(2). Let $v_1, v_2, \ldots, v_m$ be (0,1)-vectors of dimension $n$, and $m=n^{O(1)}$. Find a polynomial time algorithm that finds a (0,1)-...
14
votes
1answer
446 views

Log-space reduction from Parity-L to CNOT circuits?

Question. In their paper Improved simulation of stabilizer circuits, Aaronson and Gottesman claim that simulating a CNOT circuit is ⊕L-complete (under logspace reductions). It is clear that it ...
2
votes
1answer
110 views

Finding mapping between two spatial representations of the same objects

I have two matrices $U$ and $V$. $U$ is $n \times n$ and $V$ is $n \times m$. (Both are empirical results of an experiment.) I would like to find a linear transformation $A$, $m \times n$, such that $...
0
votes
0answers
331 views

norms of compressible and incompressible vector

Let $a$ be a vector in $R^m$, such that $\sum_{i=1}^ma_i=0$ I would like to bound $\sqrt{2m(2m-1)}\|a\|_{\infty}$ by $\sqrt{2m}\|a\|_2$ (or other way arround with the sharp constants), in the case ...
9
votes
2answers
515 views

Midpoint solutions to linear programs

There is a linear program for which I want not merely a solution but a solution that's as central as possible on the face of the polytope that assumes the minimal value. A priori, we expect the ...
-5
votes
1answer
310 views

Solving a system of linear inequations

Consider the following system of inequalities: $Ax=b$; $x\geq 0$; A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. a) How feasibility of this system can ...
3
votes
0answers
80 views

Efficiently Detecting “edges” in the time frequency plane

Given a signal $y(t)\in\mathbb{R}$ I wish to detect edge patterns. $s(f,t)$ is a time-frequency decomposition of $y(t)$ in some window $(t-n,t+n)$ so that $f$ loosely corresponds to a local frequency....
4
votes
0answers
534 views

What about apply maxplus algebra for all-pairs shortest paths?

I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
10
votes
1answer
370 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.
8
votes
0answers
132 views

Extension of Cheeger's inequality with distinguished vertices

The standard Cheeger's inequality for graph $G$ states that $\frac{1}{2}$ $\lambda$ < $\phi(G)$ < $\sqrt{2\lambda}$ where $\lambda$ is the second smallest eigenvalue of the normalized ...
15
votes
1answer
338 views

Two matrices related by a permutation $B = P A P^T$ - complexity

What is computational complexity of the following problem: given two complex $n\times n$ matrices $A$ and $B$ check if there is a permutation matrix $P$ such that: $$B = P A P^T.$$ If it helps, one ...
19
votes
2answers
856 views

Linearly independent Fourier coefficients

A basic property of vector spaces is that a vector space $V \subseteq \mathbb{F}_2^n$ of dimension $n-d$ can be characterized by $d$ linearly independent linear constraints - that is, there exist $d$ ...
14
votes
1answer
5k views

The computational complexity of matrix multiplication

I am looking for information about the computational complexity of matrix multiplication of rectangular matrices. Wikipedia states that the complexity of multiplying $A \in \mathbb{R}^{m \times n}$ by ...
10
votes
1answer
231 views

Boolean error correcting code over $\mathbb{F}_q$

Is there any known construction of a linear error correcting code $\mathsf{ECC}:\mathbb{F}_q^n \to \mathbb{F}_q^m$ (with reasonable parameters), such that when given a Boolean vector $v\in \{0,1\}^n$, ...
11
votes
1answer
344 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 ...