Substitution Rule for Matrices/Einstein Summation Convention

Summation Convention

Let $\mathbf A$ be a square matrix of order $n$.

The Substitution Rule for Matrices can be expressed using the Einstein summation convention as:

$(1): \quad \delta_{i j} a_{j k} = a_{i k}$

$(2): \quad \delta_{i j} a_{k j} = a_{k i}$

where:

$\delta_{i j}$ is the Kronecker delta

$a_{j k}$ is element $\tuple {j, k}$ of $\mathbf A$.

The index which appears twice in these expressions is the element $j$, which is the one summated over.

Sources

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