Talk:Permutation matrix

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I have been told (as the answer to an exam question) that:

For an nxn matrix A, and nxn permutation matrix P:
det(A) = det(inv(P).A.P)

Anyone care to explain why?

--KarlOkeeffe

A permutation matrix is an orthogonal matrix, which has determinant 1 or -1, and the determinant of the inverse is the inverse of the determinant of the matrix. And 1/1=1 and 1/-1=-1, so det(inv(P).A.P) = det(inv(P)) * det(A) * det(P) = either 1 * det(A) * 1 or -1 * det(A) * -1 = det(A). Or something like that. Similar matrix may or may not be relevant, if the term exists. (Suppose it's probably called that in english as well as danish.) Κσυπ Cyp 10:17, 15 Oct 2003 (UTC)

More simply, permutation matrices are always invertible (since S_n is a group) and inv(P)AP is a similarity transformation, and the determinant is a similiarity invariant. Dysprosia 11:26, 8 Mar 2005 (UTC)

Contents

[edit] Constant line sums

I have added an entry on a possible generalization of permutation matrices. Perhaps someone can clear up the following points.

  1. is there a name for my generalization of permutation matrices ?
  2. how is my generalization related to the generalization on the page Generalized_permutation_matrix ?
  3. what is the relation to double-stochastic matrices ?

MathMartin 17:13, 28 Jun 2004 (UTC)

See above for the name. Divide by c and you get a doubly stochastic matrix.Zaslav 11:16, 4 November 2006 (UTC)

[edit] Rows Or Columns?

The relation P_\pi P_\sigma=P_{\pi\circ\sigma} contradicts

P_\pi\mathbf g=\begin{bmatrix}g_{\pi(1)}\\\vdots\\g_{\pi(n)}\end{bmatrix}

(The RHS gives a right action.) The reasonable solution seems to be to use columns instead of rows in the definition, cf. de:Permutationsmatrix.--gwaihir 22:58, 15 February 2006 (UTC)

[edit] Left and Right Prodruct

The product PM, permutes the rows of M. I corrected that in the text. Please check it.

It's correct.Zaslav 11:14, 4 November 2006 (UTC)

[edit] Incoherent section

The section "Solving for P" makes no sense. What are A and B? Certainly, not arbitrary matrices. I suggest this section be removed. If not, it must be completely rewritten so that the reader can tell what it is about. (I cannot.) Zaslav 11:13, 4 November 2006 (UTC)

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