Determinants of operators and matrices II Let V be a finite dimensional C vector space and let T be an operator on V Recall The characteristic polynomial fT is Y fT t t d tn a1 tn 1 an where d dim GE det T Y d 1 n an Example 1 Cyclic permutations Let B v1 vn and let T be the operator sending v1 to v2 v2 to v3 and so on but then vn to v1 Then the characterisictic polynomial of T is fT t tn 1 and det T 1 n 1 In this example our linear transformation just permutes the basis Our next step is to discuss more general cases of this 1 Permutations Definition 2 A permutation of the set 1 n is a bijective function from the set 1 n to itself Equivalently it is a list 1 n such that each element of 1 n occurs exactly once The set of all permutations of length n is denoted by Sn Examples in S5 2 3 4 5 1 cycle of length 5 2 4 3 1 5 cycle of length 3 2 4 5 1 3 cycle of length 3 and disjoint cycle of length 2 Definition 3 The sign of a permutation is 1 m where m is the number of pairs i j where 1 i j n but i j Here s an easy way to count Arrange 1 2 n in one row and again in a row underneath 1 2 n in a row below Draw lines connecting i in the first row to i in the second Then m is the number of crosses Examples 1 2 3 4 5 so m 4 and sgn 1 1 2 2 3 3 1 4 4 5 5 so m 4 and sgn 1 1 2 3 2 3 1 4 5 4 5 so m 5 and sgn 1 2 3 1 4 5 Theorem 4 If and are elemnts of Sn and is their composition then sgn sgn sgn 2 Examples Let s just look at what happens to one typical pair There are really four possibilities 1 2 sgn 1 2 1 sgn 1 1 2 1 2 sgn 1 1 2 sgn 1 1 2 1 2 sgn 1 1 2 1 sgn 1 2 3 1 2 sgn 1 2 1 1 sgn 1 2 A more complicated example 1 2 3 4 5 so m 4 and sgn 1 3 2 1 5 4 so m 5 and sgn 1 1 1 2 3 4 5 2 3 4 5 1 2 3 so m 5 and sgn 1 4 5 Example 5 A cycle of length n has n 1 crossings and so its sign is 1 n 1 Note that this is the same as the determinant of the corresponding linear transfomration 4 Definition 6 Let A be an n n matrix Then X det A sgn a1 1 a2 2 an n Sn Example 7 When n 2 there are two permuations and we get det A a1 1 a2 2 a1 2 a2 1 In general there are n permuations in Sn a very big number It is often useful to think of a matrix A as a bunch of columns if A is a matrix let Aj be its jth column Then we can think of det as a function of n columns instead of a function of matrices det A det A1 A2 An Theorem 8 Let A and B be n n matrices Q 1 If A is upper triangular det A i ai i 2 det A is a linear function of each column when all the other columns are fixed and similarly for the rows 3 If A0 is obtained from A by interchanging two columns then det A0 det A 4 More generally if A0 is obtained from A by a permutation of the columns then det A0 sgn det A 5 If two columns of A are equal det A 0 6 det AB det A det B 7 det At det A Here are some explanations 1 If A is uppertriangular aij 0 if j i Now if Sn is not the identity i i for some i and then ai i 0 Thus the only term is the sum det A X is when id 2 This is fairly clear if you think about it Imagine if a01j ca1j for all j for example 3 Suppose for example that A0 is obtained from A by interchanging the first two columns Let be the permutation interchanging 1 and 2 Then for any j a0i j ai j 5 and for any a0i i ai i det A0 X X X sgn a01 1 a02 2 a0n n sgn a1 1 a2 2 an n sgn a1 1 a2 2 an n det A 4 Is proved in exactly the same way 5 Follows from 3 since then det A det A 6 Recall that in fact Bj b1 j e1 bn j en where ei is the jth standard basis vector for F n written as a column Recall also that if A and B are matrices then the jth column of AB which we write as AB j is X X X AB j ABj A bi j ei bi j Aei bi j Ai i i i So det AB det AB ABn X 1 AB2 X X det bi 1 Ai bi 2 Ai bi n Ai i i i Using the fact that det is linear with respect to the columns over and over again we can multiply this out X det AB b 1 1 b 2 2 b n n det A 1 A 2 A n where here the sum is over all functions from the set 1 n to itself But by 5 the determinant is zero if is not a permutation and if it is we just get the determinant of A times the sign of So miracle we end up with X det AB b 1 1 b 2 2 b n n sgn det A det B det A 7 det A X sgn a1 1 a2 2 an n X X sgn 1 a1 1 1 a2 1 2 an 1 n sgn a 1 1 a 2 2 a n n det At 6