Einstein notation
Abstract description

The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.

In physics, a scalar is invariant under transformations of basis. In particular, a Lorentz scalar is invariant under a Lorentz transformation. The individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.

As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.

The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form e = e ⊗ e. Any tensor T in V ⊗ V can be written as:

V *, the dual of V, has a basis e, e,..., e which obeys the rule

where δ is the Kronecker delta. As

the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
combination:two_responses