Have you seen this “exponent of matrices” before?

Question

No, not that matrix exponential.

Given an $n \times m$ matrix $A$ and an $n \times k$ matrix’s $B$ define their “funky exponent” $B^A$, an $m \times k$ matrix, entrywise by $$(B^A)_{ij} := \prod_{z = 1}^n B_{zj}^{A_{zi}}$$

I hope I didn’t mess up the order of the indices here. Have you seen this matrix before, and if so could you point me to somewhere to read about it?

The funky exponent satisfies a few nice properties:

• $B^{A + C} = B^A \odot B^C$, where $\odot$ is the Hadamard entrywise product, and
• $(B \odot C)^A = B^A \odot C^A$, and
• $A^{BC} = (A^B)^C$.

I figured it might be called the “Hadamard exponent” on account of the first two properties, but I couldn’t find anything with that name.