# Questions tagged [linear-algebra]

For challenges involving linear algebra, the mathematics of vector spaces and linear mappings between them.

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### NxM List Combination Closest to Target

Given a list of N lists, each containing M positive integers, and a separate list of M positive integers (target values), return a list of N scalars (integers with a value of 0 or more) that ...
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### Appease the Picky Eater

Your friend Jack is a picky eater. He only likes certain foods, and he only likes to eat a certain amount of them each day. Despite this, Jack has very strict calorie and macronutrient goals that he ...
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### Linear dependences over the field with two elements

The $d$-dimensional vector space $\mathbb{F}_2^d$ over the field with two elements $\mathbb{F}_2$ is the set of vectors of $d$ bits. Addition of vectors is bitwise xor. A linear dependence is ...
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### Explore a Klarner-Rado sequence [duplicate]

One of the Klarner-Rado sequences is defined as follows: the first term is $1$ for all subsequent terms, the following rule applies: if $x$ is present, so are $2x+1$ and $3x+1$ the sequence ...
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### Reorder a matrix, twice

You are given a square $n \times n$ matrix $A$, and a list (or vector) $u$ of length $n$ containing the numbers $1$ through $n$ (or $0$ through $n-1$). Your task is to reorder the ...
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### Billiard balls collision

Given the 2-dimensional positions and velocities of a pair of billiard balls right before impact, calculate their velocities after a perfectly elastic collision. The balls are assumed to be ideal ...
378 views

### Decompose Commutators

A theorem in this paper1 states that every integral n-by-n matrix M over the integers with trace M = 0 is a commutator, that means there are two integral matrices A,B of the same size as M such that M ...
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### Find the inverse of a 3 by 3 matrix

Challenge Given nine numbers, a, b, c, d, e, f, g, h, i, as input which correspond to the square matrix: \mathbf{M} = \begin{pmatrix}a& b& c\\ d& e&...
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### Print the positive non-zero integer n-tuple(s) that solve an inequality within a bound

Suppose I have a linear inequality like x0A0 + x1A1 + ... + xnAn <= C with xi a non-zero positive integer and Ai and C a positive non-zero multiple of 0.01. ...
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### Calculate the Hafnian as quickly as possible

The challenge is to write the fastest code possible for computing the Hafnian of a matrix. The Hafnian of a symmetric 2n-by-2n ... 3k views

### Codegolf the Hafnian

The challenge is to write codegolf for the Hafnian of a matrix. The Hafnian of an 2n-by-2n symmetric matrix ... 1k views

### Find the dot product of Rationals

I was at a friend's house for dinner and they suggested the idea of a "Prime-factor vector space". In this space the positive integers are expressed as a vector such that the nth element in the ...
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### Generalized matrix trace

Inspiration. Given (by any means): A two-argument (or single argument consisting of a two-element list) black box function, f: ℤ+ × ℤ+ → ℤ+ (input and output are 1,...
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### Characteristic polynomial

The characteristic polynomial of a square matrix $A$ is defined as the polynomial $p_A(x) = \det(Ix-A)$ where $I$ is the identity matrix and $\det$ the determinant. Note that this definition ...
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### Eigenvalues of a Matrix

Given a square matrix, output the matrix's eigenvalues. Each eigenvalue should be repeated a number of times equal to its algebraic multiplicity. The eigenvalues of a matrix ... 606 views

### Solve a matrix equation by Jacobi's method (Revised)

Mathematical Background Let A be an N by N matrix of real numbers, b a vector of N real numbers and x a vector N unknown real numbers. A matrix equation is Ax = b. Jacobi's method is as follows: ...
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### Determinant of an Integer Matrix

Given a square integer matrix as input, output the determinant of the matrix. Rules You may assume that all elements in the matrix, the determinant of the matrix, and the total number of elements in ... 2k views

### Is the matrix rank-one?

Given a matrix of integers, test if it's rank-one, meaning that every row is a multiple of the same vector. For example, in ...
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### Verify Eigenpairs

In this challenge, you will be given a square matrix A, a vector v, and a scalar λ. You will ...
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### Symbolic matrix multiplication

There are lots of different ways to explain matrix multiplication. I'll stick with a single figure since I believe most people here are familiar with it (and the figure is very descriptive). If you ...
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### Arnold's Cat Map

Challenge Given a colour raster image* with the same width and height, output the image transformed under Arnold's cat map. (*details see below) Definition Given the size of the image ...
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### Totally Invertible Submatrices

(inspired by this question over on Math) The Definitions Given an n x n square matrix A, we can call it invertible if there ...
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### Generate binary matrices which are distinct up to reflections

Here are all the 2x2 binary matrices ...
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### Dot Product of Diagonals

This challenge is very simple. You are given as input a square matrix, represented in any sane way, and you have to output the dot product of the diagonals of the matrix. The diagonals in specific are ...
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### Recursive 2x2 determinant

The determinant of a 2 by 2 matrix a b c d is given by ad - bc. Given a matrix of digits with dimensions 2n by 2n, n ≥ 1, ...
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### Find the Cross Product

The cross product of two three-dimensional vectors $\vec a$ and $\vec b$ is the unique vector $\vec c$ such that: ﻿$\vec c$ is orthogonal to both $\vec a$ and $\vec b$ The magnitude of \\$...
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### Construct the Identity Matrix

The challenge is very simple. Given an integer input n, output the n x n identity matrix. The identity matrix is one that has <...
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### Cofactor Matrices

The cofactor matrix is the transpose of the Adjugate Matrix. The elements of this matrix are the cofactors of the original matrix. The cofactor (i.e. the element of the cofactor matrix at row i and ...
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### Construct a companion matrix

You have a number of polynomials who are lonely, so make them some companions (who won’t threaten to stab)! For a polynomial of degree n, there is an ...
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### Reduced Row-Echelon Form of a Matrix

The goal of this challenge is to create a program that takes in a matrix and outputs its reduced row-echelon form. A matrix is in reduced row-echelon form if it meets all of the following ...
612 views

### Multiply Pauli Matrices

The Pauli matrices are a set of 2x2 matrices which appear very commonly in quantum physics (no, you don't need to know any quantum physics for this challenge). If we include the identity in the set, ...