Einstein Notation - Common Operations in This Notation

Common Operations in This Notation

In Einstein notation, the usual element reference for the mth row and nth column of matrix A becomes . We can then write the following operations in Einstein notation as follows.

Inner product (hence also Vector dot product)

Using an orthogonal basis, the inner product is the sum of corresponding components multiplied together:

This can also be calculated by multiplying the covector on the vector.

Vector cross product

Again using an orthogonal basis (in 3d) the cross product intrisically involves summations over permutations of components:

where

and is the Levi-Civita symbol.

Matrix multiplication

The matrix product of two matrices and is:

equivalent to

Trace

For a square matrix, the trace is the sum of the diagonal elements, hence the sum over a common index .

Outer product

The outer product of the column vector by the row vector yields an m×n matrix A:

Since i and j represent two different indices, there is no summation and the indices are not eliminated by the multiplication.