The discussion revolves around calculating the square of a permutation matrix, specifically using the example of the permutation represented by the matrix (4,2,5,6,3,1) over the set (1,2,3,4,5,6). Participants explore different methods for computing (sigma)^2 and clarify the relationship between permutation matrices and their inverses.
Participants express differing views on the methods for calculating (sigma)^2 and the relationship to finding the inverse. There is no consensus on the best approach, and some confusion remains regarding the methods discussed.
Participants highlight potential misunderstandings in the methods for calculating permutation squares and inverses, indicating that the approaches may not be straightforward and depend on careful mapping of elements.
sigma=mfb said:You can do it element by element. For example, 4 gets mapped to 6. What does 6 get mapped to? That will give you one entry in your matrix for sigma^2.
Alternatively, do it cycle by cycle with a similar approach.
No, 1->6 is a part of sigma2. This specific example happens to have some mappings common with its inverse because sigma3 keeps several elements unchanged, but that is pure coincidence.ilyas.h said:This is the wrong approach because by doing this, you're calculating (sigma)^-1, that's the inverse.
mfb said:No, 1->6 is a part of sigma2. This specific example happens to have some mappings common with its inverse because sigma3 keeps several elements unchanged, but that is pure coincidence.
Swap the two lines and sort the columns to make the first line "1 2 3 4 5 6" again.ilyas.h said:how would you find the inverse of the permutation then?
mfb said:Swap the two lines and sort the columns to make the first line "1 2 3 4 5 6" again.
This is equivalent to "for 1, look where it appears in the bottom line, find which elements gets mapped to 1, map 1 to this element" and so on.