# Definition:Matrix/Summation Convention

## Contents

## Definition

The **summation convention** is a notational device used in the manipulation of matrices, in particular square matrices in the context of physics and applied mathematics.

If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed.

Thus the summation sign can be omitted, and expressions can be written more compactly.

## Examples

### Trace of Matrix

The trace of $A$, using the summation convention, is:

- $\tr \paren A = a_{ii}$

### Substitution Rule

The Substitution Rule for Matrices can be expressed using the 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 $\left({j, k}\right)$ of $\mathbf A$.

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

### Determinant of Order 3

The determinant of a square matrix of order $3$ $\mathbf A$ can be expressed using the summation convention as:

- $\det \left({\mathbf A}\right) = \dfrac 1 6 \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right) a_{i r} a_{j s} a_{k t}$

Note that there are six indices which appear twice, and so six summations are assumed.

### Matrix Product

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