Let x,y in RR^n. Let A be a n x xn positive-definite symmetric matrix. Is there a general formula for x^T Ax * y^T Ay?

vedentst9i

vedentst9i

Answered question

2022-11-04

Let x , y R n . Let A be a n × n positive-definite symmetric matrix. Is there a general formula for x T A x y T A y?
For example, let x = [ 2 2 ] , y = [ 3 3 ] , A = [ 1 0 0 1 ]
Then x T A x y T A y = 8 18 = 144
This is equal to 2 ( x y ) T A ( x y ) where ( x y ) = [ 6 6 ] is the Hadamard product of x,y.
It seems that x T A x y T A y = 2 ( x y ) T A ( x y ) also works for x = [ 5 5 ] , y = [ 7 7 ] . Is this true in the general case, and if so, how do I prove it? If not, how can I find and prove a general formula?

Answer & Explanation

hamputlnf

hamputlnf

Beginner2022-11-05Added 12 answers

The Hadamard product won't help, but the Kronecker product distributes over the matrix product, so one can write
( x T A x ) ( y T A y ) = ( x T A x ) ( y T A y ) = ( x y ) T ( A A ) ( x y )
To start things off, the product between the two scalar expressions can also be replaced by a Kronecker product.

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