This is taken from one of the lecture slides, where a simplification of the quadratic formula is att

Govindennz34j

Govindennz34j

Answered question

2022-05-10

This is taken from one of the lecture slides, where a simplification of the quadratic formula is attempted:
( x x A ) T A ( x x A ) + a T ( x x a ) + b
and from there I expand the brackets, which yields:
( x x A ) T A ( x x A ) + a T ( x x a ) + b = x T A x x A T A x x A T A x A + a T x a T x a + b
In the next step the factorization is used to rearrange the formula:
( x x A ) T A ( x x A ) + a T ( x x a ) + b = x T A x + ( A T x A x A T A T + a ) T x + ( x A T A x A + b a T x a )
I do not understand how the linear term has been transposed from ( x A T A T ) T x. Can someone explain this ?

Answer & Explanation

cegielnikmzjkf

cegielnikmzjkf

Beginner2022-05-11Added 14 answers

What about this simplification?
Let x ¯ = [ x 1 ] , then we have that
x x A = [ 1 x A ] x ¯ , x x a = [ 1 x a ] x ¯ , a = [ 0 a ] x ¯ ,   a n d b = x ¯ T [ 0 0 0 b ] x ¯ .
x ¯ T ( [ 1 x A T ] A [ 1 x A ] + 1 2 [ 0 a T ] [ 1 x a ] + 1 2 [ 1 x a T ] [ 0 a ] + [ 0 0 0 b ] ) x ¯ .
If we expand that we get
x ¯ T [ A A x A + a / 2 x A T A x A a T x a + b ] x ¯ .

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