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 ...
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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 ...
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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 ...
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 ...
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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 ...
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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 ...
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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 ...
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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}),\\ .....\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...