Given two matrices $A$ of size $n \times m$ and $B$ of size $m \times n$, it is well known that the matrix multiplication between $A$ and $B$ is defined by
$$(AB)_{ij} = \sum_k A_{ik} B_{kj}$$ of size $n \times n$;
and the Kronecker product between $A$ and $B$ is defined by
$$A \otimes B = \left[\begin{array}{{c}*6}
A_{11} B_{11} & A_{11} B_{12} & \cdots & A_{11} B_{1n} & A_{12} B_{11} & \cdots & A_{1m} B_{1n} \\
A_{21} B_{11} & A_{21} B_{12} & \cdots & A_{21} B_{1n} & A_{22} B_{11} & \cdots & A_{2m} B_{1n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
A_{n1} B_{m1} & A_{n1} B_{m2} & \cdots & A_{n1} B_{mn} & A_{n2} B_{m1} & \cdots & A_{nm} B_{mn}
\end{array}\right]$$ of size $mn \times mn$.
Q: Is there a specific notation for the following concept that is between matrix multiplication and Kronecker product?
$C = f(A, B)$ of size $n \times n \times m$ s.t.
$$C_{ijk} = A_{ik}B_{kj}.$$
We have
$$(AB)_{ij} = \sum_k C_{ijk}$$
and $C$ can be extracted from $A \otimes B$.
Best Answer
The Kronecker delta symbol is defined such that $$ \def\vd{\vec\delta} \delta_{ij} = \begin{cases} 1\quad {\rm if}\;i=j \\ 0\quad {\rm otherwise}\\ \end{cases} $$ Extending this to three indices yields the generalized symbol $$\vd_{ijk} = \begin{cases} 1\quad {\rm if}\;i=j=k \\ 0\quad {\rm otherwise}\\ \end{cases} $$ This third-order tensor can be used to write the desired product $$ \def\C{{\cal C}} \C = A\cdot\vd\cdot B \quad\implies\quad \C_{ijk} = \sum_{p=1}^m\sum_{q=1}^m A_{ip}\,\vd_{pjq}\,B_{qk} $$