1.10.5 Einstein Summation Notation

Einstein Summation Notation is a concise mathematical notation used in tensor algebra to simplify the representation of sums over indices.

Einstein Summation Notation is a shorthand convention, introduced by Albert Einstein, in which an index that appears exactly twice within a single term, once as an upper index and once as a lower index, is automatically understood to be summed over its full range of values, without the need to write an explicit summation symbol. This convention removes the visual clutter of repeated Σ symbols from tensor expressions and, more importantly, makes the structure of an expression, which indices are summed and which remain free, immediately apparent from the placement of the indices themselves.

The convention applies specifically to the pairing of one upper and one lower index sharing the same label; two upper indices sharing a label, or two lower indices sharing a label, are not summed under the standard convention and instead typically signal an error unless a metric-based exception is explicitly invoked. This asymmetric rule, favoring an upper-lower pairing, is what ties the summation convention directly to the transformation properties that distinguish contravariant from covariant tensor components.

The Core Rule

Automatic Summation Over Repeated Indices

Under the Einstein convention, an expression such as a_i b^i is automatically interpreted as a sum over all values of i from 1 to n, without any explicit summation sign, whereas in ordinary summation notation the same quantity would require writing the sum symbol explicitly.

a_{i} b^{i} \equiv \sum_{i = 1}^{n} a_{i} b^{i}

Why Upper-Lower Pairing Is Required

The convention is restricted to pairs consisting of one upper and one lower index because such a pairing is precisely what is guaranteed to produce a basis-independent result: a contravariant index transforms with the Jacobian, a covariant index transforms with the inverse Jacobian, and summing over a matched upper-lower pair causes these two transformation factors to cancel exactly, leaving the summed quantity invariant under a change of coordinates.

a_{i'} b^{i'} = \frac{\partial x^{j}}{\partial x^{i'}} \frac{\partial x^{i'}}{\partial x^{k}} a_{j} b^{k} = a_{j} b^{j}

Free Indices and Their Role

What Remains Unsummed

An index that appears only once within a term is a free index, and it is not summed; it remains as a live index in the result, indicating that the expression still depends on that index and represents an entire family of components rather than a single number. The set of free indices in an expression determines the type of tensor the expression produces.

c^{i} = A_{j}^{i} b^{j} free index i, dummy index j

Matching Free Indices Across an Equation

For a tensor equation to be valid under the summation convention, every free index appearing on one side of the equation must appear, with the same name and the same upper or lower placement, on the other side. This requirement functions as an immediate consistency check: any mismatch reveals an error before the equation is evaluated numerically.

Worked Illustrations

Matrix-Vector Multiplication

The familiar formula for multiplying a matrix by a vector is written, under the summation convention, as a single compact expression, with the shared index j summed automatically and the free index i indicating that the result is itself a vector.

c^{i} = A_{j}^{i} x^{j}

Matrix Multiplication

Multiplying two matrices together is written similarly, with the inner shared index summed, leaving two free indices, i and k, that determine the row and column of the resulting matrix.

C_{k}^{i} = A_{j}^{i} B_{k}^{j}

The Trace

The trace of a matrix, ordinarily written as an explicit sum over diagonal entries, becomes a single term with both indices identified and no free indices remaining, under the summation convention.

tr (A) = A_{i}^{i}

Exceptions and Extensions

Repeated Indices of the Same Type

When two indices of the same variance, both upper or both lower, need to be compared or combined without triggering the standard summation convention, this is typically indicated by explicit notation or by stating that no summation is implied, since the default convention only governs matched upper-lower pairs.

Use With the Metric to Raise and Lower

The metric tensor is itself governed by the same summation convention when used to raise or lower an index, contracting one of its own indices against the index being converted, which allows index-raising and index-lowering operations to be written with the same compact summation notation used throughout the rest of tensor algebra.

v_{i} = g_{ij} v^{j}

Why the Convention Matters

Compactness Without Loss of Precision

The Einstein summation convention allows expressions involving many nested sums to be written with no summation symbols at all, without sacrificing any precision, since the rule for determining which indices are summed, repetition in matched upper-lower position, is unambiguous and mechanically applied. This combination of brevity and precision is why the convention has become the standard default throughout index-based tensor algebra.

A Foundation for Reading Tensor Expressions

Because so much of tensor notation depends on correctly identifying which indices are free and which are summed, fluency with the Einstein summation convention is a prerequisite for reading and manipulating essentially any expression written in index notation, whether component-based or abstract, making it one of the most load-bearing conventions in the entire notational apparatus of tensor algebra.