Try to figure out how to find the derivate of a double summation. The equation goes like: (del)/(del x_i) (x^(TT) A x) where x=((x_1),(vdots),(x_n)) \text{ and } A=((a_11,cdots,a_(1n)),(vdots,ddots,vdots),(a_(n1),cdots,a_(n n)))

skystasvs

skystasvs

Answered question

2022-09-12

Tryto figure out how to find the derivate of a double summation. The equation goes like: x i ( x A x ) where
x = ( x 1 x n ) and A = ( a 11 a 1 n a n 1 a n n ) .

Answer & Explanation

vermieterbx

vermieterbx

Beginner2022-09-13Added 14 answers

A matrix can be decomposed into its symmetric and skew parts
A ± = 1 2 ( A ± A T ) A = A + + A
and the value of your double sum is unchanged if A is replaced with A +
x T A x = x T A + x
Now consider a function which uses different vectors on the left and right
ϕ = x T A + y = ( A + y ) T x ϕ x = A + y
And since A + is symmetric
ϕ = y T A + x = ( A + x ) T y ϕ y = A + x
If y is a function of x then
ϕ x = ( A + y ) ( x x ) + ( A + x ) ( y x )
Finally, setting y=x yields
( x T A + x ) x = 2 A + x = ( A + A T ) x

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