Lecture #2 Notes

# xk1 12 trace and determinant 1210 denition for

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online