Talk:Permutation matrix

message: 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)
replies: text: 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)
thread_title:  Determinant of perm matrix 

message: I have added an entry on a possible generalization of permutation matrices. Perhaps someone can clear up the following points.
#is there a name for my generalization of permutation matrices ?
#how is my generalization related to the generalization on the page Generalized_permutation_matrix ?
#what is the relation to double-stochastic matrices ?
MathMartin 17:13, 28 Jun 2004 (UTC)
replies: text: See above for the name.  Divide by c and you get a doubly stochastic matrix.Zaslav 11:16, 4 November 2006 (UTC)
thread_title: Constant line sums

message: 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)
thread_title:  Rows Or Columns? 

message: 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)
thread_title:  Left and Right Prodruct 

message: Shouldn't a part be added to say that the multiplication of P on the right of a matrix M (instead of multiplying P by M, as in left multiplication, multiply M by P), will cause a permutation of the columns of M, in an analogous manner as it permutes the rows in left multiplication? I think a proof of both should also be included. If no one posts a proof or an explanation, I will do so myself here in the talk page, and then it can be edited.Goldencako 15:49, 14 November 2007 (UTC)
thread_title:  Right Multiply 

message: Could anyone who happens to have some time & the wisdom provide a graphical representation somewhere on this?  E.g., the "projection matrix page" has a great example of what the matrix is actually doing, "in real life".  For instance, a 2-D projection matrix can project a 3-D object onto that 2-D plane.
I have some vague sense that a permutation matrix is changing the direction of a vector (or matrix) without changing its magnitude, just from looking at the math (and it somehow seems "obvious" to me, to abuse a term).  But since the permutation matrix itself is an orthogonal matrix, I am also getting the sense that it is changing the direction of a vector (or matrix) in some way that is orthogonal to the original.  But I cannot get my head around what that "looks" like; I cannot get an intuitive sense of what's going on, and therefore cannot imagine how I'd use this future applications.
--Mike Williamson  —Preceding undated comment added 02:40, 27 December 2011 (UTC)
thread_title:  Graphical Representation? 

message: \pi  = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 \\ 1 & 3 & 0 & 2 & 4 \end{pmatrix} = (0132)(4)	\pi  = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 \\ 1 & 3 & 0 & 2 & 4 \end{pmatrix} = (0132)(4)	
                                                                                       	                                                                                       	
Left action (prefix)M\bigl(\pi(i), i\bigr) = 1                                         	Right action (postfix)M\bigl(i, \pi(i)\bigr) = 1                                       	
                                                                                       	                                                                                       	
                                                                                       	                                                                                       	
There are two ways to assign a matrix to a permutation, corresponding to the two ways of multiplying permutations.
There are two ways to draw arrows in the chosen matrix, one similar to two-line and the other to cycle notation.	
There are two ways to assign a matrix to a permutation, corresponding to the two ways of multiplying permutations.
There are two ways to draw arrows in the chosen matrix, one similar to two-line and the other to cycle notation.	
There are two ways to assign a matrix to a permutation, corresponding to the two ways of multiplying permutations.
There are two ways to draw arrows in the chosen matrix, one similar to two-line and the other to cycle notation.
The table on the right was introduced by me, and then removed ("hid massive undecipherable table"), brought back and removed again ("hid unhelpful diagram") by Wcherowi.
I agree that the table can be a bit confusing, because it shows both ways of multiplication, but this confusion comes from reality and not from the table. I think it is necessary to show an example for both ways to assign a matrix to a permutation.
One could make the table clearer by removing the lower row with the cycles. But I think it is a good idea to show, that both two line and cycle notation can be seen in a permutation matrix. One could also make it clearer by adding labels on the left as in v:Permutation notation.
If I change the images from 0-based to 1-based permutations, the examples in the article could use the same permutation. Watchduck (quack) 13:27, 26 January 2017 (UTC)
replies: text: To be clear about the editing history, I had introduced an error while attempting to correct the text and so I decided to self-revert and put the page back to its earlier form. This diagram was just an innocent bystander in this process. My original reasons for removing the diagram still stand. First of all, it is much too large; totally dominating the initial view of this article and making a point which is only a minor wrinkle in the subject. The diagram is not well integrated with the text of the article nor with the general literature on the subject. The major problem, however, is that unless the reader is quite familiar with the subject, these arrows passing through a stylized matrix do not convey much meaning. Although they make for a pretty picture a reader would be baffled by their intent. The concept involved has a much simpler explanation that I hope I have given in the text. The same problem exists with the diagrams appearing later in the article, but they might be saved by some appropriate text and I will need some time to consider how to do this. --Bill Cherowitzo  (talk) 18:45, 26 January 2017 (UTC)
thread_title: Example images

message: In the definition section, when say:
The m \times m permutation matrix P_\pi = (p_{ij}) obtained by permuting the columns of the identity matrix I_m, that is, for each i, p_{ij} = 1 if j = \pi(i) and 0 otherwise ...
Could be, in my opinion 
The m \times m permutation matrix P_\pi = (p_{ij}) obtained by permuting the columns of the identity matrix I_m, that is, for each i, p_{ij} = 1 if j = \pi(i) and p_{ij}=0 otherwise ...
— Preceding unsigned comment added by Nikocof (talk • contribs) 
: Sure, that is clearer.  I've made the change (in two places). --JBL (talk) 13:06, 17 March 2020 (UTC)
thread_title:  Definition 

message: Users @JayBeeEll and @XOR'easter make the ridiculous claim that adding a sentence which notes that the permutation matrices can be precisely characterized as as the non-negative orthogonal matrices somehow magically "buries" and "obscures" other information in the article. Can these users explain their justification for this ridiculous claim? 2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 19:55, 5 August 2022 (UTC)
replies: text: I'm not going to repeat the several full sentences I wrote about this in edit summaries, since you've clearly read them -- if there's something specifically you don't understand about them, let me know.  It is totally possible that this characterization could be placed somewhere in the article (not in the place you put it), if you have an appropriate source for it.  Do you? JBL (talk) 20:00, 5 August 2022 (UTC)
replies: text: @JayBeeEll Your edit summaries were merely false assertions backed by absolutely nothing: no reasoning and no justification. And the place I put it was a good one because it was the place in the article where orthogonal matrices were mentioned. 2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 20:03, 5 August 2022 (UTC)
replies: text: The fact that you don't understand something does not make it false.  By the way, I've edited your addition (now in a more plausibly appropriate location) so that it is actually a true statement.  Do you have a source for it? JBL (talk) 20:05, 5 August 2022 (UTC)
replies: text: @JayBeeEll What exactly is it you claim I "don't understand"? Also, your edit makes no sense: Permutation and orthogonal matrices are real matrices, by definition. So there was nothing false about my statement. It seems like it's you who needs to review your definitions. I will add a source. 2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 20:09, 5 August 2022 (UTC)
replies: text: The orthogonal group can be defined over any field. JBL (talk) 20:14, 5 August 2022 (UTC)
replies: text: @JayBeeEll I said orthogonal matrices, not orthogonal group, did I not? 2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 20:23, 5 August 2022 (UTC)
replies: text: An orthogonal matrix is an element of an orthogonal group; they can be defined over any field.  (If you had followed the link I included and read the introductory section, you would have learned this!)  Of course more people study O_n(R) than any other orthogonal group, but that doesn't make the other ones not exist. JBL (talk) 20:28, 5 August 2022 (UTC)
replies: text: @JayBeeEll I expected you to reply with these ridiculous mental gymnastics. Orthogonal matrix, without qualification, simpliciter, refers to real matrices (and the same is true for "orthogonal group"). That's why, e.g., unitary matrices and unitary group are referred to by their own name. I can pull up various textbooks to support this, but it would be an exercise in futility since you already know I'm right.  2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 20:40, 5 August 2022 (UTC)

text: What you do not understand (apparently -- though I think you probably are perfectly capable of understanding it, once you get over your pique about being reverted) is that taking a sentence about topic X with a straightforward logical structure, splitting it into two sentences, and then inserting a separate sentence about topic Y into the middle of the two, obscures and obfuscates the original point X.  Since the fact X (that permutation matrices are invertible, and their inverses are equal to their transposes) is really important, this is a bad thing to do. JBL (talk) 20:19, 5 August 2022 (UTC)
replies: text: @JayBeeEll What really "piques" me is that you decided to unilaterally expunge the information from the article completely instead of rewording or moving it if necessary, which is what would be implied by your edit summary. 2601:547:500:6940:D11A:50DC:8EB8:EDC0 (talk) 20:49, 5 August 2022 (UTC)

text:  That would be great, thank you in advance. JBL (talk) 20:20, 5 August 2022 (UTC)
thread_title:  Permutation matrices are the non-negative orthogonal matrices