inner -- compute the inner product of two tensors.

Usage:

c = inner(f,g)

c = f @ g
Inputs:
- f, an instance of the type NCRingElement,
- g, an instance of the type NCRingElement,

Description

The inner product of two words $\mathtt{w}$ and $\mathtt{v}$ in a free associative algebra is defined as $1$ if $\mathtt{w} = \mathtt{v}$ and $0$ otherwise. This extends bilinearly to an inner product on the whole associative algebra.

The inner product can be used to access the coefficient of a tensor over a single word.

i1 : R = wordAlgebra(3);

i2 : t = 2*[1,2,3]_R + [2,3,1]_R + 4*[3,3,3,3]_R; t //wordFormat

o3 = 4 [3, 3, 3, 3] + [2, 3, 1] + 2 [1, 2, 3]

i4 : [1,2,3]_R @ t

o4 = 2

o4 : QQ

i5 : [3,3,3,3]_R @ t

o5 = 4

o5 : QQ

i6 : ([1,2,3]_R @ t) == (t @ [1,2,3]_R)

o6 = true

It is also a convenient way to evaluate linear combinations of tensor entries:

i7 : A = CAxisTensor(3,R);

i8 : vol = sgnVolTensor(R);

i9 : A @ vol -- the signed volume of the canonical axis path in 3 dimensions.

     1
o9 = -
     6

o9 : QQ

Ways to use inner:

inner(NCRingElement,NCRingElement)

For the programmer

The object inner is a method function.

The source of this document is in PathSignatures/documentation.m2:918:0.