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Unformatted text preview: Lecture 11 Interlude: Theorem 1. Det A ( H ) = Tr ( e AH ) Proof. It is enough to see both sides agree when H is a standard basis matrix E ij . Left Hand Side: Det A ( E ij ) = d dt Det ( A + tE ij ) | t =0 = d dt ( DetA + (- 1) i + j tA ij ) | t =0 (where A ij is the minor of A) = (- 1) i + j A ij . Right Hand Side: ( BE ij ) pq = n r =1 B pr ( E ij ) rq = n r =1 B pr ir jq = B pi jq Tr ( BE ij ) = n p =1 B pi jp = B ji . In particular, Tr ( e AE ij ) = e A ji = (- 1) i + j A ij . Lemma 1. Let F : U R where U X and X is a normed space. Suppose F is differentiable at a U and has a local maximum value at a then F a ( h ) = for all h . (Exercise) 1 Proof. Suppose BWOC there exists x U such that F a ( x ) 6 = 0. WLOG we can take F a ( x ) > 0 (if F a ( x ) < 0 look at- x ). Since a is a local maximum, there exists 1 such that k y- a k < 1 f ( y ) f ( a ) . Now let F a ( x ) = s > 0....
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
- Summer '10