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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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 Fall '12
 BernhardBodmann
 Multiplication, Matrices, Counting

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