Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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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
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7 votes
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Are the minimal quantum and classical span programs the same?

A span program is a linear-algebraic way of specifying a boolean function introduced here which has found recent application in quantum query complexity. A span program for a function $f: \{0,1\}^n \...
Artem Kaznatcheev's user avatar
10 votes
1 answer

Finding a cutting plane that splits a polyhedron evenly

Say we have a polyhedron in standard form: \begin{equation*} \begin{array}{rl} \mathbf{A}\mathbf{x} = \mathbf{b} \\\\ \mathbf{x} \ge 0 \end{array} \end{equation*} Are there any known methods for ...
7 votes
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What's new in sparse eigensystems solution

As a part of other work I need to solve relatively large (~1E5x1E5) and sparse (~100 non-zero elements in each raw in few blocks) hermitian eigensystems. Usually only few eigenvalues+vectors are ...
Misha's user avatar
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19 votes
2 answers

A data structure for minimum dot product queries

Consider $\mathbb{R}^n$ equipped with the standard dot product $\langle \cdot, \cdot \rangle$ and $m$ vectors there: $v_1, v_2, \ldots, v_m$. We want to build a data structure that allows queries of ...
ilyaraz's user avatar
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3 votes
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Taking Square Roots of Matrices over Z/nZ

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a ...
WuTheFWasThat's user avatar
5 votes
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Hardness of finding eigenvalues?

Is there a setting in which finding eigenvalues/eigenvectors is computationally hard? Or at least, not known to be computationally easy? For example, how computationally hard or easy is it to find ...
WuTheFWasThat's user avatar
10 votes
2 answers

Restricting entries of unitary operators to real numbers and universal gate sets

In Bernstein and Vazirani's seminal paper "Quantum Complexity Theory", they show that a $d$-dimensional unitary transformation can be efficiently approximated by a product of what they call "near-...
Henry Yuen's user avatar
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10 votes
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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
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21 votes
1 answer

What is the most general structure on which matrix product verification can be done in $O(n^2)$ time?

In 1979, Freivalds showed that verifying matrix products over any field can be done in randomized $O(n^2)$ time. More formally, given three matrices A, B, and C, with entries from a field F, the ...
Robin Kothari's user avatar
2 votes
2 answers

LU factorization of a 0-1 matrix

I have a rather naive question on LU factorization which probably should be easy to answer. Say I have a matrix with entries only from $\{0,1\}$. When can we expect to get an LU factorization of such ...
Turbo's user avatar
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14 votes
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Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)

Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same. First, I will define a few ...
Akash Kumar's user avatar
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16 votes
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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
9 votes
2 answers

Polynomial algorithms for UPB (Unextendable Product Bases)

Consider a Hilbert space $H = H_1 \otimes \dots \otimes H_n$. An Unextendable Product Basis (UPB) is a set of product vectors $\vert v_i \rangle = \vert v_i^1 \rangle \otimes \dots \otimes \vert v_i^n ...
Marcin Kotowski's user avatar
9 votes
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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
62 votes
4 answers

Evidence that matrix multiplication can be done in quadratic time?

It is widely conjectured that $\omega$, the optimal exponent for matrix multiplication, is in fact equal to 2. My question is simple: What reasons do we have for believing that $\omega = 2$? I'm ...
Steve Flammia's user avatar
8 votes
1 answer

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
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97 votes
2 answers

What is the actual time complexity of Gaussian elimination?

In an answer to an earlier question, I mentioned the common but false belief that “Gaussian” elimination runs in $O(n^3)$ time. While it is obvious that the algorithm uses $O(n^3)$ arithmetic ...
Jeffε's user avatar
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13 votes
1 answer

Matrix multiplication in $O(n^2 \log n)$

I was searching about Matrix multiplication, So I first visit wiki matrix multiplication algorithms, In references I found a paper which claim that uses $O(n^2 log(n))$ algorithm , I'd going to read ...
Saeed's user avatar
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11 votes
2 answers

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
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5 votes
3 answers

Calculating a fast matrix vector product between vector of reals and a 0-1 matrix

Given: some vector $R=(r_1...r_l)$ - real numbers, and a set of distinct vectors with $0$ or $1$ coordinates $$\begin{array}{c} V_1=(c_{1,1} ... c_{1,l}),\\ V_2=(c_{2,1} ... c_{2,l}),\\ .....\...
user avatar
20 votes
3 answers

Survey on algorithms/complexity of linear algebra

I am looking for a good survey on algorithms and complexity of linear algebra (operations like rank, inverse, eigenvalues, ... for Boolean, $\mathbb{F}_p$, and integers/rationals matrices) with ...
Kaveh's user avatar
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2 votes
1 answer

Determinants and bilinear forms

If one calculates the product of diagonal elements of the $U$ matrix in a $LUP$ factorization of a given matrix $A$, one can calculate the determinant of $A$. Also it is known that $LUP$ factorization ...
Turbo's user avatar
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46 votes
8 answers

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
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5 votes
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Integer multiplication where regular Fourier Transform approach would fail to provide best upper bound

I have a problem where multiplication of integers via regular Fourier Transform based multiplication technique would fail to provide best upper bound since the sequences of bits in both integers are ...
Turbo's user avatar
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18 votes
2 answers

A Boolean function that is not constant on affine subspaces of large enough dimension

I'm interested in an explicit Boolean function $f \colon \\{0,1\\}^n \rightarrow \\{0,1\\}$ with the following property: if $f$ is constant on some affine subspace of $\\{0,1\\}^n$, then the dimension ...
Alexander S. Kulikov's user avatar
15 votes
1 answer

Sparse Walsh-Hadamard Transform

The Walsh-Hadamard transform (WHT) is a generalization of the Fourier transform, and is an orthogonal transformation on a vector of real or complex numbers of dimension $d = 2^m$. The transform is ...
Suresh Venkat's user avatar

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