The `inner()` function in the `stokes` package

In stokes: The Exterior Calculus

knitr::opts_chunk$set(echo = TRUE)
library("stokes")
library("emulator")
set.seed(0)

![](`r system.file("help/figures/stokes.png", package = "stokes")`){width=10%}

inner

Function inner() returns the inner product corresponding to a matrix.

The inner product of two vectors $\mathbf{x}$ and $\mathbf{y}$ is usually written $\left\langle\mathbf{x},\mathbf{y}\right\rangle$ or $\mathbf{x}\cdot\mathbf{y}$, but the most general form would be $\mathbf{x}^TM\mathbf{y}$ where $M$ is a matrix. Noting that inner products are multilinear, that is $\left\langle\mathbf{x},a\mathbf{y}+b\mathbf{z}\right\rangle=a\left\langle\mathbf{x},\mathbf{y}\right\rangle + b\left\langle\mathbf{x},\mathbf{z}\right\rangle$ and $\left\langle a\mathbf{x} + b\mathbf{y},\mathbf{z}\right\rangle=a\left\langle\mathbf{x},\mathbf{z}\right\rangle + b\left\langle\mathbf{y},\mathbf{z}\right\rangle$ we see that the inner product is indeed a multilinear map, that is, a tensor.

We can start with the simplest inner product, the identity matrix:

inner(diag(7))

Note how the rows of the tensor appear in arbitrary order. Verify:

x <- rnorm(7)
y <- rnorm(7)
V <- cbind(x,y)
c(as.function(inner(diag(7)))(V),sum(x*y))  # should match

A more stringent test would be to use a general matrix:

M <- matrix(rnorm(49),7,7)
f <- as.function(inner(M))
c(f(V),quad.3form(M,x,y)) # should match

(function emulator::quad.3form() evaluates matrix products efficiently; quad.form(M,x,y) returns $x^TMy$). Of course, we would expect inner() to be a homomorphism:

M1 <- matrix(rnorm(49),7,7)
M2 <- matrix(rnorm(49),7,7)
g <- as.function(inner(M1+M2))
c(g(V),quad.3form(M1+M2,x,y)) # should match

Now, if the matrix is symmetric the corresponding inner product should also be symmetric:

h <- as.function(inner(M1 + t(M1)))
c(h(V), h(V[,2:1]))  # should match

Also positive-definite matrix should return a positive quadratic form:

M3 <- crossprod(matrix(rnorm(56),8,7))  # 7x7 pos-def matrix
as.function(inner(M3))(kronecker(rnorm(7),t(c(1,1))))>0  # should be TRUE

Alternating forms

The inner product on an antisymmetric matrix should be alternating:

jj <- matrix(rpois(49,lambda=3.2),7,7)
M <- jj-t(jj) # M is antisymmetric
f <- as.function(inner(M))
c(f(V),f(V[,2:1])) # differ in sign

stokes documentation built on Jan. 18, 2022, 1:11 a.m.