Definition:Matrix Product (Conventional)/Einstein Summation Convention

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$.

The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the Einstein summation convention as:

Then:

$c_{i j} := a_{i k} \circ b_{k j}$

The index which appears twice in the expressions on the right hand side is the entry $k$, which is the one summated over.

Sources

• 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.2$: The summation convention