What is the name of this simple operation between tensor and matrix? In my study, I faced with the following formula and considered shortening the description.
Let $A$ be an $q \times m \times p$ tensor including $m \times p$ matrices $A_0, A_1, \ldots, A_{q-1}$, and $B$ be an $p \times n$ matrix.
We have another $q \times m \times n$ tensor such that
$$
C = \begin{bmatrix} A_0 B & A_1 B & \cdots & A_{q-1} B\end{bmatrix}.
$$
Then, how can we write $C$ by $A$ and $B$?
It is a simple notation C = A @ B in Python/NumPy.
What is the name of this operation?
 A: This operation is an example of a trace, or a contraction (of indices), of the tensor product $A \otimes B$. In general, you can form a trace for each choice of pair (covariant index, contravariant index) of indices on a particular vector space $\Bbb V$. If you have an inner product on $\Bbb V$ you can identify the two index types with one another and hence contract on any pair of indices on $\Bbb V$.
In our situation there is essentially only one trace. Borrowing the choice of covariant and contravariant indices specified in the comments realizes our operation as the contraction $$\operatorname{tr}: (\Bbb A^* \otimes \Bbb B^* \otimes \Bbb C) \times (\Bbb C^* \otimes \Bbb D) \to \Bbb A^* \otimes \Bbb B^* \otimes \Bbb D$$ for vector spaces $\Bbb A, \Bbb B, \Bbb C, \Bbb D$, of respective dimensions $q$, $m$, $p$, $n$, characterized by its definition on pairs $$((\alpha \otimes \beta \otimes c), (\gamma \otimes d)) \mapsto \gamma(c) \alpha \otimes \beta \otimes d .$$
I suspect that NumPy essentially treats tensors as elements of tensor products of $\Bbb R^k$ for various $k$, each equipped with the standard inner product, in which case we can view our contraction map as
$$\operatorname{tr}: (\Bbb R^q \otimes \Bbb R^m \otimes \Bbb R^p) \times (\Bbb R^p \otimes \Bbb R^n) \to \Bbb R^q \otimes \Bbb R^m \otimes \Bbb R^n.$$
In situations where there is more than one possible trace of the tensor product, one sometimes sees the notation $\operatorname{tr}_{ab}$ to indicate the contraction of the $a$th and $b$th indices (using a background inner product if necessary), so we might denote our trace as $\operatorname{tr}_{34}$.
