Questions tagged [permutation-matrices]

A permutation matrix is a square matrix that has exactly one $1$ in each row and each column and $0$s elsewhere. Permutation matrices are orthogonal matrices.

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The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from assignment 4. Prove that the transpose of a…
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Can you completely permute the elements of a matrix by applying permutation matrices?

Suppose I have a $n\times n$ matrix $A$. Can I, by using only pre- and post-multiplication by permutation matrices, permute all the elements of $A$? That is, there should be no binding conditions, like $a_{11}$ will always be to the left of…
Landon Carter
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What are the properties of eigenvalues of permutation matrices?

Up till now, the only things I was able to come up/prove are the following properties: $\prod\lambda_i = \pm 1$ $ 0 \leq \sum \lambda_i \leq n$, where $n$ is the size of the matrix eigenvalues of the permutation matrix lie on the unit circle I am…
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If permutation matrices are conjugate in $\operatorname{GL}(n,\mathbb{F})$ are the corresponding permutations conjugate in the symmetric group?

There is a standard embedding of the symmetric group $S_n$ into $\operatorname{GL}(n,\mathbb{F})$ (for any field $\mathbb{F}$) that sends each permutation in $S_n$ to the corresponding permutation matrix. As this is a group homomorphism, certainly,…
James
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Row swap changing sign of determinant

I was wondering if someone could help me clarify something regarding the effect of swapping two rows on the sign of the determinant. I know that if $A$ is an $n\times n$ matrix and $B$ is an $n\times n$ matrix obtained from $A$ by swapping two rows,…
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Span of permutation matrices

The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, I am interested in a description of a subset of…
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LU decomposition; do permutation matrices commute?

I have an assignment for my Numerical Methods class to write a function that finds the PA=LU decomposition for a given matrix A and returns P, L, and U. Nevermind the coding problems for a moment; there is a major mathematical problem I'm having…
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Characterizing sums of permutation matrices

Given an $n$ by $n$ matrix $A$ whose rows and columns sum to $m \in \mathbb N$ and entries are nonnegative integers, does there exist a permutation matrix $P$ such that $A - P$ has only nonnegative entries? If this is true, then we can write $A$ as…
user28877
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1 answer

Special permutations of $\{1,2,3,\ldots,n\}$

How do you show that number of permutations of $\{1,2,3,\ldots,n\}$ such that image of no two consecutive numbers is consecutive is $$n! + \sum_{k = 1}^{n}(-1)^k\sum_{i = 1}^{k}\dbinom{k - 1}{i - 1}\dbinom{n - k}{i}2^i(n - k)!$$ In short we need to…
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How do you find all solutions to the matrix equation $XAX=A^T$?

I was recently asked to solve a problem in a programming interview involving word squares, and on further reflection I realized it could be recast as a linear algebra question. Since my solution has a worst-case time complexity of $O(n!)$ if $n$ is…
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Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial.

I was told that Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial (where the conjugacy is considered only in $S_{n}$). The first implication is clear to me i.e. permutation matrices representing…
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Problem involving permutation matrices from Michael Artin's book.

Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is $$ P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{bmatrix} $$ This is the matrix that permutes the…
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Number of permutations of matrices with unique rows and columns

Consider an $m$ by $n$ matrix filled with integers in $\left[0, b\right[$. There would be $b^{mn}$ possible matrices. Two matrices would be considered equivalent (in this system) iff you can switch some rows and columns in the matrix and they are…
Artyer
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What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?

According to Birkhoff, $n$-by-$n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as a convex combination of finitely many…
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What is the dimension of the span over $\mathbb{F}$ of permutation matrices on $n$ elements if $char(\mathbb{F})$ divides $n$?

I'm trying to generalize Birkhoff's theorem and prove that over any field $\mathbb{F}$ any matrix whose sum of rows and columns is equal to some constant, is a linear combination of permutation matrices. For fields where the characteristic $p$ does…
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