Index Notation and Summation Conventions

Index notation labels components of vectors, covectors, matrices, tensors, basis vectors, tensor factors, and summation variables. It is a compression scheme for linear algebra. Used carefully, it makes matrix products, contractions, angular-momentum identities, and tensor-product bookkeeping transparent; used casually, it hides convention errors.

The site default is conservative: sums are usually written explicitly unless a page declares that Einstein summation convention is being used. This page explains both styles and the translation between them.

Indices Label Components

After a basis $\{e_i\}$ is chosen, a vector $v$ can be written

$$v = \sum_i v^i e_i.$$

The index $i$ labels the component attached to $e_i$. It is not a physical observable by itself; it is a coordinate label relative to a basis.

A covector, or dual vector, may be written

$$\omega = \sum_i \omega_i e^i,$$

where $\{e^i\}$ is the dual basis satisfying

$$e^i(e_j) = \delta^i{}_j.$$

The placement of indices can carry meaning. In this finite-dimensional language, upper indices often denote vector components and lower indices often denote covector components. Many quantum pages use simpler matrix notation when no distinction is needed.

Explicit Sums

The safest convention is to write sums explicitly:

$$(Av)^i = \sum_j A^i{}_j v^j.$$

Here $A^i{}_j$ is the matrix component that sends input component $j$ into output component $i$. The repeated $j$ is summed, while $i$ remains as the free output index.

For matrix multiplication,

$$(AB)^i{}_k = \sum_j A^i{}_j B^j{}_k.$$

This says that $B$ acts first on the input index $k$, then $A$ maps the intermediate index $j$ to the output index $i$.

Einstein Summation Convention

Einstein summation convention suppresses the summation sign when an index is repeated once upstairs and once downstairs:

$$(Av)^i = A^i{}_j v^j.$$

The repeated index $j$ is summed. The free index $i$ is not summed. The free indices must match on both sides of an equation.

Thus

$$w^i=A^i{}_j v^j$$

is shorthand for

$$w^i = \sum_j A^i{}_j v^j.$$

Einstein convention is efficient, but only after it has been declared. Do not assume it in a page that has not opted into it.

Free, Dummy, and Repeated Indices

A free index labels the component of the object being defined. In

$$w^i=A^i{}_j v^j,$$

$i$ is free and $j$ is dummy. The dummy index can be renamed:

$$A^i{}_j v^j = A^i{}_k v^k.$$

The expression

$$A^i{}_j v^k$$

has free indices $i,j,k$ unless a summation convention or explicit sum says otherwise. It is not the same kind of object as $w^i$.

As a rule of thumb, an index should not appear more than twice in one term under Einstein convention. If it does, rewrite the expression with explicit sums.

Kronecker Delta

The Kronecker delta is the identity matrix in index notation:

$$\delta^i{}_j = \begin{cases} 1, & i=j,\\ 0, & i\ne j. \end{cases}$$

It leaves components unchanged:

$$\delta^i{}_j v^j = v^i.$$

It also collapses explicit sums:

$$\sum_j \delta^i{}_j v^j = v^i.$$

In an orthonormal basis, the inner product of basis vectors is often written

$$\langle e_i\vert e_j\rangle = \delta_{ij}.$$

The same symbol appears in many forms because it represents the identity pairing between matching labels.

Levi-Civita Symbol

In three dimensions, the Levi-Civita symbol $\epsilon_{ijk}$ is antisymmetric in all indices and is defined by

$$\epsilon_{123}=1,$$

with sign changes under swaps of indices and zero when any two indices are equal.

In Euclidean Cartesian three-vector notation, authors often sum repeated lower spatial indices. With that convention, it encodes cross products:

$$(a\times b)_i = \epsilon_{ijk}a_j b_k$$

with repeated lower spatial indices summed. It also appears in angular-momentum and Pauli-matrix identities, for example

$$[\sigma_i,\sigma_j] = 2i\,\epsilon_{ijk}\sigma_k.$$

The Levi-Civita symbol is a convention-heavy object. Sign errors often come from swapping index order, mixing orientation conventions, or silently changing whether indices start from $1$ or from $0$.

Matrix Components and Dirac Notation

In an orthonormal basis $\{\lvert e_i\rangle\}$, an operator $A$ has components

$$A_{ij} = \langle e_i\vert A\vert e_j\rangle.$$

If

$$A\lvert e_j\rangle = \sum_i A_{ij}\lvert e_i\rangle,$$

then $j$ is the input column index and $i$ is the output row index. With a state

$$\lvert\psi\rangle = \sum_j c_j\lvert e_j\rangle,$$

the components of $A\lvert\psi\rangle$ are

$$(Ac)_i = \sum_j A_{ij}c_j.$$

This is the same rule as matrix multiplication. Dirac notation emphasizes the operator and basis vectors; index notation emphasizes the component labels.

Tensor-Product and Multi-Indices

For a bipartite basis $\{\lvert i\rangle_A\otimes\lvert \alpha\rangle_B\}$, a state may be written

$$\lvert\psi\rangle = \sum_{i,\alpha} c_{i\alpha} \lvert i\rangle_A\otimes\lvert\alpha\rangle_B.$$

The pair $(i,\alpha)$ is a multi-index: it records both subsystem labels. A local operator $A\otimes I_B$ acts as

$$\left( A\otimes I_B \right)_{i\alpha,j\beta} = A_{ij}\delta_{\alpha\beta}.$$

This identity is a useful antidote to missing identity factors. The $A$ indices act on subsystem $A$, while the Kronecker delta carries subsystem $B$ through unchanged.

Worked Example

Let

$$A = \begin{pmatrix} 1 & 2\\ 0 & -1 \end{pmatrix}, \qquad v = \begin{pmatrix} 3\\ 4 \end{pmatrix}.$$

Using explicit sums,

$$(Av)_1 = \sum_{j=1}^2 A_{1j}v_j = A_{11}v_1+A_{12}v_2 = 1\cdot3+2\cdot4 = 11,$$

and

$$(Av)_2 = \sum_{j=1}^2 A_{2j}v_j = 0\cdot3+(-1)\cdot4 = -4.$$

With Einstein convention, the same calculation is simply

$$w_i=A_{ij}v_j, \qquad w_1=11, \qquad w_2=-4,$$

after declaring that repeated indices are summed.

Common Mistakes

• Assuming repeated indices are summed when the page has not declared Einstein convention.
• Leaving different free indices on the two sides of an equation.
• Using the same dummy index for two unrelated sums in a way that creates ambiguity.
• Forgetting that $A_{ij}$ usually means output row $i$, input column $j$.
• Treating Kronecker deltas as decorative rather than as identity maps over specified labels.
• Swapping Levi-Civita indices and losing a minus sign.
• Suppressing tensor-product identity factors before subsystem ordering is clear.

Cross-Links

• Mathematical Notation Used in This Volume
• Bases and Coordinates
• Matrices as Linear Maps
• Dirac Notation as Linear Algebra
• Tensor Products
• Angular Momentum Algebra
• Representation Translation Table
• Symbol Map

References

• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Translate $w^i=A^i{}_j v^j$ into an explicit sum.

Solution

Assuming Einstein summation convention,

$$w^i = \sum_j A^i{}_j v^j.$$

The index $i$ is free, while $j$ is summed.

2. Simplify $\delta^i{}_j\delta^j{}_k v^k$ using Einstein convention.

Solution

First contract over $j$:

$$\delta^i{}_j\delta^j{}_k = \delta^i{}_k.$$

Then contract over $k$:

$$\delta^i{}_k v^k = v^i.$$

3. In three dimensions, what is $\epsilon_{213}$?

Solution

The index order $(2,1,3)$ is obtained from $(1,2,3)$ by one swap, so

$$\epsilon_{213}=-1.$$