xk1 12 trace and determinant 1210 denition for

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Unformatted text preview: equal to the span of zj ’s. Thus we normalize zk = yk ￿y k ￿ such that {z1 , z2 , . . . , zk } is orthonormal and has the same span as {x1 , x2 , . . . , xk−1 }. 1.2 Trace and Determinant 1.2.10 Definition. For A ∈ Mn , where the entries of A are represented by A = (ai,j )n =1 , we i,j let n ￿ tr[A] = aj,j j =1 and det[A] = ￿ sgn(σ )a1,σ(1) a2,σ(2) . . . an,σ(n) σ ∈ Sn where the sum runs over all n! permutations σ of the n items {1, 2, . . . n}, and sgn(σ ) is 1 if σ is an even permutation and −1 if it is odd. 1.2.11 Remark. • If A = [a1 |a2 | . . . |an ], then det(A) = f (a1 , a2 , . . . , an ) and it is the only function f which is linear in each column vector, alternating in the columns, f (a1 , a2 , . . . , aj , . . . , ak , . . . , an ) = −f (a1 , a2 , . . . , ak , . . . , aj , . . . , an ) and it is normalized, so that det(I ) = 1. • If C...
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