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 = 144This 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?

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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  =  144This 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?

hamputlnf · Accepted Answer

The Hadamard product won&#039;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.

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?

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