I'm not sure how to describe the multiplication of off-diagonal elements of a (co)variance matrix (covariances) with corresponding elements of it's correlation matrix, in a succinct way.
In full, I want to describe the multiplication of elements of a correlation matrix with the square root of the corresponding elements of the covariance matrix. But it is specifying the corresponding elements of the respective matrices that I'm struggling with.
How can I describe this with correct notation?
Thanks
Best Answer
If $A, B\in M_n(\mathbb R)$ are given then write $$A\circ B$$ for their entrywise product: the entry at $(i,j)$ of $A\circ B$ is $\,a_{ij}b_{ij}\,$. This is called the Schur product of $A$ and $B$. And also: the Hadamard product.
It has the important property that $A\circ B\,$ is a positive matrix if $A$ and $B$ are positive.
You have commutativity $B\circ A = A\circ B$, in sharp contrast to matrix multiplication.
It may conveniently be applied to select or weight individual entries, which is your aim.
Consider the example $$\pmatrix{1&0&0\\ 0&1&0\\ 0&0&1}\:\circ\: \pmatrix{j&e&s\\m&a&th\\j&o&y} \;=\; \pmatrix{j&0&0\\0&a&0\\0&0&y}$$ selecting the diagonal entries.