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What is the permutation matrix?

What is the permutation matrix?

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere.

How does permutation matrix work?

A permutation matrix is a square matrix obtained from the same size identity matrix by a permutation of rows. Such a matrix is always row equivalent to an identity.

Is the identity matrix A permutation matrix?

Theorem. An identity matrix is an example of a permutation matrix.

What is the determinant of permutation matrix?

The determinant of a permutation matrix is either 1 or –1, because after changing rows around (which changes the sign of the determinant) a permutation matrix becomes I, whose determinant is one. Definition: the sign of a permutation, sgn(σ), is the determinant of the corresponding permutation matrix.

What is permutation matrix in LU decomposition?

LU factorization is a way of decomposing a matrix A into an upper triangular matrix U , a lower triangular matrix L , and a permutation matrix P such that PA = LU . These matrices describe the steps needed to perform Gaussian elimination on the matrix until it is in reduced row echelon form.

What is a permutation matrix example?

A permutation matrix is a matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to . Every row and column therefore contains precisely a single 1 with 0s everywhere else, and every permutation corresponds to a unique permutation matrix.

Is permutation matrix a orthogonal matrix?

A permutation matrix is an orthogonal matrix, that is, its transpose is equal to its inverse.

What are the eigenvalues of permutation matrix?

The only eigenvalues of permutation matrices are 1 and -1 by theorem 2, so we only need to consider the eigenvectors of eigenvalue 1 and -1. By lemma 4, eigenvalue 1 contributes the number of eigenvectors equal to the number of cycles, and by lemma 6, eigenvalue -1 contributes the number of even length cycles.

What is the purpose of LU factorization?

LU decomposition is a better way to implement Gauss elimination, especially for repeated solving a number of equations with the same left-hand side. That is, for solving the equation Ax = b with different values of b for the same A.

What is the principle of LU factorization method?

An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U, A=LU.

Are permutation matrices orthogonal?