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
Famous quotes containing the words common and/or operations:
“Mankind’s common instinct for reality ... has always held the world to be essentially a theatre for heroism. In heroism, we feel, life’s supreme mystery is hidden. We tolerate no one who has no capacity whatever for it in any direction. On the other hand, no matter what a man’s frailties otherwise may be, if he be willing to risk death, and still more if he suffer it heroically, in the service he has chosen, the fact consecrates him forever.”
—William James (1842–1910)
“A sociosphere of contact, control, persuasion and dissuasion, of exhibitions of inhibitions in massive or homeopathic doses...: this is obscenity. All structures turned inside out and exhibited, all operations rendered visible. In America this goes all the way from the bewildering network of aerial telephone and electric wires ... to the concrete multiplication of all the bodily functions in the home, the litany of ingredients on the tiniest can of food, the exhibition of income or IQ.”
—Jean Baudrillard (b. 1929)