TITLE: Is it possible to write the Hadamard product of two matrices in tensor notation?
QUESTION [5 upvotes]: Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common component, i.e. tensor, notation? Would it be something like $A^{\alpha \beta} B^{\alpha \beta}$ or is that not sufficient? Would this lead to conflicts with Einsteins summation convention?

REPLY [4 votes]: If you aren't using the summation convention then $C^{\alpha\beta}=A^{\alpha\beta}B^{\alpha\beta}$ is fine.
If you are using the summation convention then $A^{\alpha\beta}B^{\alpha\beta}$ means
$$\sum_{\alpha\beta}A^{\alpha\beta}B^{\alpha\beta}$$
which is a scalar rather than a matrix. In this case the thing to do is to define a new tensor $\delta^\alpha_{\;\beta\gamma}$ such that
$$\delta^\alpha_{\;\beta\gamma}=\begin{cases}
1&\text{if}\;\alpha=\beta=\gamma\\
0&\text{otherwise}\\
\end{cases}$$
in your basis. Then define $C^{\alpha\beta}=\delta^\alpha_{\;\gamma\delta}\delta^\beta_{\;\eta\phi}A^{\gamma\eta}B^{\delta\phi}$.