Determinants of operators and matrices Let V be a finite dimensional C vector space and let T be an operator on V Recall The characteristic polynomial fT is the polynomial Y fT t t d where d dim GE The Cayley Hamilton theorem fT T 0 as an operator on V Goal compute fT without actually computing the generalized eigenspaces or even the eigenvalues just from MB T for any arbtirary basis B for V Y fT t t d tn a1 tn 1 an for some list of complex numbers ai Definition 1 If T L V and fT t tn a1 tn 1 an is its characteristic polynomial then trace T a1 and det T 1 n an Last time If A MB T then trace T trace A 1 X ai i Outline 1 Define the determinant of a matrix det A 2 Check that if A is uppertriangular det A Y aii 3 Show that if A and B are n n matrices det AB det A det B 4 Conclude that det S 1 AS det A for every invertible S 5 Conclude that if A MB T then det A det T Step 1 is probably the hardest How to find the definition Many approaches I ll follow the book more or less Let s look at cyclic spaces Theorem 2 Suppose that V is T cyclic so that there is a v V with V span v T v T 2 v T n 1 v Then v T v T n 1 v is a basis of V and T n v c0 v c1 T v cn 1 T n 1 v for a unique list c0 cn 1 in C Then the characteristic polynomial of T is p t tn cn 1 tn 1 c1 t c0 Example 3 Last time we looked at T x1 x2 x2 x1 Let v 1 0 Then v T v is a basis for V and T 2 v v Thus c0 1 and c1 0 the p t t2 1 Proof The equation for T n v says that p T v 0 It follows that p T T i v T i p T v 0 for all i and since the T i v s span V p T 0 Since the list v T v T n 1 v is indepenent there is no polynomial of smaller degree that annihilates T Thus p is the minimal polynomial of T and since its degree is the dimension of V p is also the characteristic polyonmial of T Corollary 4 In the cyclic case above det T 1 n 1 c0 Example 5 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 2 Permutations Definition 6 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 2 4 3 1 5 2 4 5 1 3 The first of these is cycle of length 5 Note that the second doesn t move 3 or 5 and can be viewed as a cyclic permutation of the set 1 2 4 The last permuation can be viewed as a the product composition of a cyclic permutation of 1 2 4 and a cyclic permutation of 3 5 Definition 7 The sign of a permuation 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 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 2 3 4 5 1 1 2 3 4 5 so m 4 and sgn 1 2 4 3 1 5 1 2 3 4 5 so m 5 and sgn 1 2 4 5 1 3 3 Example 8 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 d Theorem 9 If Sn then sgn sgn sgn Omit the proof at least for now Definition 10 Let A be an n n matrix Then X det A sgn a1 1 a2 2 an n Sn Example 11 When n 2 there are two permuations and we get det A a1 1 a2 2 a1 2 a2 1 Proposition 12 Let A and B be n n matrices Q 1 If A is upper triangular det A i ai i 2 If B is obtained from A by interchanging two columns then det B det A 3 If two columns of A are equal det A 0 4 det A is a linear function of each column when all the other columns are fixed 5 det AB det A det B 4