matrices - MAT 312/AMS 351 Notes and Exercises on...

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MAT 312/AMS 351 Notes and Exercises on Permutations and Matrices. We can represent a permutation π S ( n ) by a matrix M π in the following useful way. If π ( i ) = j , then M π has a 1 in column i and row j ; the entries are 0 otherwise. This M π permutes the unit column vectors e 1 , e 2 , . . . , e n , by matrix multiplication, just the way π permutes 1 , 2 , . . . , n . Example. Suppose n = 6 and π = (1542)(36). Following the rule, we get M π = 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 . We can check: π (1) = 5, and M π ( e 1 ) = 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 = 0 0 0 0 1 0 = e 5 , etc. Proposition 1. For σ, π S ( n ), we have M πσ = M π M σ ; i.e. the matrix corresponding to a composition of permutations is the product of the individual matrices. Proof. On the one hand, M πσ ( e i ) = e πσ ( i ) = e π ( σ ( i )) . On the other hand, M π M σ ( e i ) = M π ( e σ ( i ) ) = e π ( σ ( i )) also. square To proceed we need some facts about determinants. (1) Every square matrix M has a determinant det M , which is a sum of products of entries in M . So if M has integer entries, det M will be an integer, etc. (2) The determinant of a 1 × 1 matrix ( a 11 ) is the number a 11 itself.
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