I've been trying to find an answer to this for some time now, if anyone can provide a definitive answer, they'll have the satisfaction of knowing that several University mathematics professors haven't been able to come up with one... I am looking for a way to transform from a Hadamard product (element-wise matrix multiplication) into a classic matrix product. Before I get flamed for posing a supposedly stupid question, the answer is related to quantum networks - (see - http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_927_0/AIF_1999__49_3_927_0.pdf" [Broken] p. 951). So more specifically - Let us start with a matrix Z, Z is an element of Real. Element wise multiplication is given by .*, regular multiplication is * . A ranges between 0 and positive infinity, B ranges between -infinity and positive infinity. A.*Z = B*Z Is there a function that maps A to B? To provide more information, think about B. If we exponentiate it to 0, we get the identity matrix (I) . If we exponentiate it by -1 (assuming it does have an inverse) then we get Z. On the left hand side of the equation - if A is uniform (i.e. all matrix elements are identical), then we get Z. If A is the identity matrix, then we get (I). I'm chiefly interested in what happens in the inbetween states (i.e. between no change and orthogonalization). Transitioning B is straightforward (exponentiate to p where p ranges between 0 and -1). Transitioning A is less so. I get the feeling that the answer has something to do with rotation, but it's n-dimensional... Any advice (or references) would be appreciated. Thanks!