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.
Read more about this topic: Einstein Notation
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