DG

Multi-index notation

... by doing things 'right', one can avoid what Elie Cartan has called the 'debauch of indices' - Mike Spivak, 1970

Multi-indexing is a collection of notational devices, whose main goal is to avoid drowning in a sea of indices.
A single multi-index is used to denote dependence of several indices: I = (i₁, ... i_n)

Definition: Given an indexing set S (typically S = Z⁺), a multi-index is an element of Sⁿ.

We can use multi-index notation to vectorize quantities: u_I = (u_i₁,... u_{i_n}),
and contract them with an operation, e.g. tensor: $\big_tensor$ u_I = u_i₁ $\tensor$ ... u_{i_n}

Summing over a multi-index: Indexed operations, such as sums, products, etc.
can be written compactly, e.g. $\sum$ _I c_I = $\sum$ _{i₁, ... i_n} c_{i₁, ... i_n}

Example: Elementary symmetric functions of a₁,...a_n are $\sum$ _{I C(k, n)} $\prod$ a_I,
where C(k,n) is the set of all increasing k-tuples in {1,...n},
e.g. if n=3, E₂(a₁,a₂,a₃) = a₁a₂+a₁a₃+a₂a₃.

Last updated: Jun 17 18:53 / Last fetched: Tue Dec 2 07:48:33 CST 2008