# Definition:Einstein Summation Convention

## Definition

The **Einstein summation convention** is a notational device used in the manipulation of matrices and vectors, 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 Einstein summation convention, is:

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

### Substitution Rule

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.

### Determinant of Order 3

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

- $\map \det {\mathbf A} = \dfrac 1 6 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$

Note that there are $6$ indices which appear twice, and so $6$ summations are assumed.

### Matrix Product

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.

### Component Form of Vector

Let $\mathbf a$ be a vector quantity.

$\mathbf a$ can be expressed in component form using the Einstein summation convention as:

- $\mathbf a = a_i \mathbf e_i$

### Dot Product

Let $\mathbf a$ and $\mathbf b$ be vector quantities.

The dot product of $\mathbf a$ and $\mathbf b$ can be expressed using the Einstein summation convention as:

\(\ds \mathbf a \cdot \mathbf b\) | \(:=\) | \(\ds a_i b_j \delta_{i j}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds a_i b_i\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds a_j b_j\) |

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

## Also known as

Some sources do not credit Albert Einstein with this notation, merely referring to it as **the summation convention**.

## Source of Name

This entry was named for Albert Einstein.

## Historical Note

When Albert Einstein was working with summations in the context of vectors and tensors, he noticed that the subscript or superscript to denote the index appeared twice.

So he had the idea of omitting the summation signs and instead using, for example, $x_i y_i$ to mean summation over the repeated subscripts over the entire applicable range.

Einstein himself humorously referred to this as:

*... a great discovery in mathematics ...*

A different (possibly apocryphal) story relates that this convention came about through a printer setting the type noticed the redundancy and suggested its omission to Einstein.

## Sources

- 1980: A.J.M. Spencer:
*Continuum Mechanics*... (previous) ... (next): $2.2$: The summation convention - 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product