Let&nbsp;x,y∈Rn. Let A be a&nbsp;n×n&nbsp;positive-definite symmetric matrix. Is there a general formula for&nbsp;xTAx⋅yTAy?For example, let&nbsp;x=22,&nbsp;y=33,&nbsp;A=1001Then&nbsp;xTAx⋅yTAy=8⋅18=144This is equal to&nbsp;2(x∘y)TA(x∘y)&nbsp;where&nbsp;(x∘y)=66&nbsp;is the Hadamard product of x,y.It seems that&nbsp;xTAx⋅yTAy=2(x∘y)TA(x∘y)&nbsp;also works for&nbsp;x=55,&nbsp;y=77. 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&amp;nbsp;x,y∈Rn. Let A be a&amp;nbsp;n×n&amp;nbsp;positive-definite symmetric matrix. Is there a general formula for&amp;nbsp;xTAx⋅yTAy?For example, let&amp;nbsp;x=22,&amp;nbsp;y=33,&amp;nbsp;A=1001Then&amp;nbsp;xTAx⋅yTAy=8⋅18=144This is equal to&amp;nbsp;2(x∘y)TA(x∘y)&amp;nbsp;where&amp;nbsp;(x∘y)=66&amp;nbsp;is the Hadamard product of x,y.It seems that&amp;nbsp;xTAx⋅yTAy=2(x∘y)TA(x∘y)&amp;nbsp;also works for&amp;nbsp;x=55,&amp;nbsp;y=77. 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?

Cody Johns · Accepted Answer

The Hadamard product won&#039;t help, but the Kronecker product distributes over the matrix product, so one can write(xTAx)(yTAy)=(xTAx)⊗(yTAy)&amp;nbsp;=(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 \(\displaystyle{x},{y}\in{\mathbb{{{R}}}}^{{{n}}}\). Let A be a

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