I have a problem about a Hadamard product (i.e. elementwise multiplication) between two terms, each term being the matrix dot product between an $n\times n$ sized matrix $M$ and the $n\times 1$ vector $a$ (term 1) and $n\times 1 $ vector $b$ (term 2):
$$(M\cdot a)\circ(M\cdot b)$$
Is there any way to re-express this equation so that it follows something along the lines of?:
$$ (M\circ M)\cdot(a\circ b) $$
I know that if $a$ and $b$ are written as diagonalized $n\times n$ matrices the re-expression works as written above, (i.e. if $A = diag(a)$ and $B = diag(b)$ then $(M\cdot A)\circ(M\cdot B) = (M\circ M)\cdot(A\circ B)$) but can't find a solution for the case where they are $n\times 1$ vectors...
I'm trying to re-express the equation to get decoupling between the matrix operator $M$ and the two vector terms $a$ and $b$. There are no constraints on the form of the re-expression aside from the attempt to combine the action of the matrix operator onto a product of both vectors.
Thank you!
The re-expression of:
$$(M\cdot a)\circ(M\cdot b)$$
With the desired effect of decoupling the matrix product from the two vectors $a$ and $b$ is:
$$(M\bullet M)\cdot(a\otimes b)$$
Where $\bullet$ is the face-splitting product (also known as the transposed Khatri-rao product) and $\otimes$ is the Kronecker product.