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Given matrices $A$ and $B$, the Hadamard product $A\circ B$ is given by $(A\circ B)_{ij}=A_{ij}\cdot B_{ij}$ (if you ask an ordinary middle schooler, this would be the "most natural" definition of matrix multiplication, haha.)

Does anyone know if it is possible to represent $A\circ B$ as a combination of other widely used matrix operations, such as addition, multiplication, taking determinant and taking inverse?

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No. All of those operations behave well with respect to conjugation, and the Hadamard product doesn't. –  Qiaochu Yuan Aug 6 '12 at 2:33
    
One should add that the matrices are of the same size, in order for the definition to apply. Also, if they are just numbers, then this is the ordinary multiplication... –  awllower Aug 6 '12 at 2:52
    
Per chance, restricted to some subgroups of the matrix group, this representation is possible?? Such as the Heisenberg group?? –  awllower Aug 6 '12 at 2:55
    
@awllower: certainly the diagonal matrices... –  Qiaochu Yuan Aug 6 '12 at 3:16
    
@QiaochuYuan: Indeed, thanks. –  awllower Aug 6 '12 at 3:21

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