Alt {stokes}

Alternating multilinear forms

Description

Converts a \(k\)-tensor to alternating form

Usage

Alt(S,give_kform)

Arguments

S

A multilinear form, an object of class ktensor

give_kform

Boolean, with default FALSE meaning to return an alternating \(k\)-tensor [that is, an object of class ktensor that happens to be alternating] and TRUE meaning to return a \(k\)-form [that is, an object of class kform]

Details

Given a \(k\)-tensor \(T\), we have

\[\mathrm{Alt}(T)\left(v_1,\ldots,v_k\right)= \frac{1}{k!}\sum_{\sigma\in S_k}\mathrm{sgn}(\sigma)\cdot T\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right) \]

Thus for example if \(k=3\):

\[\mathrm{Alt}(T)\left(v_1,v_2,v_3\right)= \frac{1}{6}\left(\begin{array}{c} +T\left(v_1,v_2,v_3\right)\quad -T\left(v_1,v_3,v_2\right)\cr -T\left(v_2,v_1,v_3\right)\quad +T\left(v_2,v_3,v_1\right)\cr +T\left(v_3,v_1,v_2\right)\quad -T\left(v_3,v_2,v_1\right) \end{array} \right) \]

and it is reasonably easy to see that \(\mathrm{Alt}(T)\) is alternating, in the sense that

\[\mathrm{Alt}(T)\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)= -\mathrm{Alt}(T)\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right) \]

Function Alt() is intended to take and return an object of class ktensor; but if given a kform object, it just returns its argument unchanged.

A short vignette is provided with the package: type vignette("Alt") at the commandline.

Value

Returns an alternating \(k\)-tensor. To work with \(k\)-forms, which are a much more efficient representation of alternating tensors, use as.kform().

Author(s)

Robin K. S. Hankin

See Also

kform

Examples

(X <- ktensor(spray(rbind(1:3),6)))
Alt(X)
Alt(X,give_kform=TRUE)

S <- as.ktensor(expand.grid(1:3,1:3),rnorm(9))
S
Alt(S)

issmall(Alt(S) - Alt(Alt(S)))  # should be TRUE; Alt() is idempotent

a <- rtensor()
V <- matrix(rnorm(21),ncol=3)
LHS <- as.function(Alt(a))(V)
RHS <- as.function(Alt(a,give_kform=TRUE))(V)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)