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Questions tagged [matrices]

For questions about construction and modification of matrices, objects represented by 2-dimensional arrays that are used to define linear operators within linear algebra.

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Automated optimization of 0-1 matrix vector multiplication

Question: Is there established procedure or theory for generating code that efficiently applies a matrix-vector multiplication, when the matrix is dense and filled with only zeros and ones? Ideally, the optimized code would make systematic use of…
Nick Alger
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Common idea in Karatsuba, Gauss and Strassen multiplication

The identities used in multiplication algorithms by Karatsuba (integers) Gauss (complex numbers) Strassen (matrices) seem very closely related. Is there a common abstract framework/generalization?
sdcvvc
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Can you multiply complex 2x2 matrices in fewer than 21 real multiplies?

It is well known that 2x2 matrices can be multiplied using just 7 (instead of the obvious 8) multiplications in the ground field (Strassen-Winograd, etc.). It is also well known that complex numbers can be multiplied using just 3 real…
Steve Linton
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Fastest way to solve a system of linear equations

I have to solve a system of up to 10000 equations with 10000 unknowns as fast as possible (preferably within a few seconds). I know that Gaussian elimination is too slow for that, so what algorithm is suitable for this task? All coefficients and…
tmwilliamlin168
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How can I quickly judge whether matrix A is the inverse matrix of B?

How can I quickly judge whether matrix A is the inverse matrix of B? This is an exercise for the course I take. This question is given in the section of randomized algorithms. So I think its solution may be related to randomized algorithms.
t24akeru
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Are there parallel matrix exponentiation algorithms that are more efficient than sequential multiplication?

One is required to find power (positive integer) of matrix of real numbers. There are lots of efficient matrix multiplication algorithms (e.g. some parallel algorithms are Cannon's, DNS) but are there algorithms that are intended exactly for finding…
TomR
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Minimal basis for set of binary vectors using XOR

I would be surprised if this isn't a well-studied problem, but I'm not sure what else to search for at this point: you're given a set of binary $n$-vectors $S \subset \{0,1\}^n$. The problem is to find another set of binary $n$-vectors $B \subset…
Martin Ender
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Is there a O(log n) algorithm for matrix exponentiation?

Is there an algorithm to raise a matrix to the $n$th power in $O(\log n)$ time? I have been searching online, but have been unsuccessful thus far.
Miles
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Counting islands in Boolean matrices

Given an $n \times m$ Boolean matrix $\mathrm X$, let $0$ entries represent the sea and $1$ entries represent land. Define an island as vertically or horizontally (but not diagonally) adjacent $1$ entries. The original question was to count the…
pgs
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Optimizing a sum of matrix chains

Edit Jan 31: important special case is when the sums form a nested structure, search for "Hasse diagram is a tree" below Here's a practically relevant variation on matrix chain problem: Find optimal way to compute a sum over all weighted paths in a…
Yaroslav Bulatov
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Real life examples of *zero* weight edges in graphs

The meaning of edges with zero weight in a weighted graph questions me for a long time, and I even asked a related question previously. Yet, when I recently read here a question on real life example of graphs with negative weight edges I was amazed…
Matthieu Latapy
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Find an optimal ordering

I came across this problem and am struggling to find a way to approach it. Any thoughts would be greatly appreciated! Suppose we are given a matrix $\{-1, 0, 1\}^{n\ \times\ k} $, for example, $$\begin{bmatrix} 1 & 0 & 1 & 0 & -1 \\ -1 &…
haijo
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2-D peak finding complexity (MIT OCW 6.006)

In a recitation video for MIT OCW 6.006 at 43:30, Given an $m \times n$ matrix $A$ with $m$ columns and $n$ rows, the 2-D peak finding algorithm, where a peak is any value greater than or equal to it's adjacent neighbors, was described as: Note: If…
user52207
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How to prove that matrix inversion is at least as hard as matrix multiplication?

Suppose we are given a matrix $A$ over real numbers and we want to computer the inverse of matrix $A$. There are various algorithms to do so and it also turn out that we can use matrix multiplication problem to solve matrix inversion problem. My…
Complexity
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Which computational model is used to analyse the runtime of matrix multiplication algorithms?

Although I have already learned something about the asymptotic runtimes of matrix multiplication algorithms (Strassen's algorithm and similar things), I have never found any explicit and satisfactory reference to a model of computation, which is…
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