In RobinHankin/onion: Octonions and Quaternions

knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
library("onion")
set.seed(1)

knitr::include_graphics(system.file("help/figures/onion.png", package = "onion"))

Introduction

To cite the onion package in publications please use @hankin2006_onion. This short document shows how matrices with octonion or quaternion entries may be created and manipulated with the onionmat class of the onion R package.

onionmat objects

A good place to start is function romat(), which creates a simple onionmat object:

set.seed(0)
options(digits=2)
romat()

This illustrates many features of the package. An onionmat object has two slots. The first slot, x, is an onion [a vector of quaternions or octonions] and the second, M, an integer matrix which is used to store attributes such as dimensions and dimnames. The elements of M and d are in bijective correspondence; thus element [b,AR] is number 17 and this is seen to be approximately $-0.81 + 0.24i -1.43j +0.37k$. Most R idiom will work with such objects, here is a brief sample.

A <- matrix(rquat(21),7,3)  # matrix() calls onion::onionmat()
A

See above how object A has no rownames or colnames and the defaults are used. We may extract components:

A[1,]

above, the resulting object is an onion but we may retain the onionmat character using drop:

A[1,,drop=FALSE]

The extraction methods operate as expected:

Re(A)
k(A)

(above, the matrices returned are numeric). Also replacement methods work as expected:

j(A) <- -1
A

Some of the summary methods work:

sum(A)

Matrix multiplication

Matrix multiplication is implemented.

A <- matrix(rquat(21),3,7)
umbral <- state.abb[1:7]
rownames(A) <- letters[1:3]
colnames(A) <- umbral

B <- matrix(rquat(28),7,4)
rownames(B) <- umbral
colnames(B) <- c("H","He","Li","Be")

A %*% B

However, it is often preferable (but no faster in this case) to use functions such as cprod() and tcprod():

C <- matrix(rquat(14),7,2)
rownames(C) <- umbral
colnames(C) <- month.abb[1:2]
cprod(B,C)

and indeed the single-argument versions work as expected:

tcprod(A) - A %*% ht(A)

Octonions

Here I consider $3\times 3$ Hermitian octonionic matrices which are known be a special case of a Jordan algebra (see the jordan package for more systematic Jordan algebra R functionality).

x <- cprod(matrix(roct(12),4,3))
x

We see that x is Hermitian symmetric:

max(Mod(Im(x+t(x))))

[that is, the imaginary components of symmetrically placed elements are mutually conjugate]. We may verify that $3\times 3$ matrices form a Jordan algebra under the composition rule $A\circ B=(AB+BA)/2$ [juxtaposition indicating regular matrix multiplication]; the identity is

[(xy)(xx) = x(y(xx)).]

First we define the Jordan product:

`%o%` <- function(x,y){(x%*%y + y%*%x)/2}

then create a couple of random Hermitian octonionic matrices:

x <- cprod(matrix(roct(12),4,3))
y <- cprod(matrix(roct(12),4,3)) # x and y are 3-by-3 Hermitian octonionic matrices

We first verify numerically that the Jordan product of two Hermitian symmetric matrices is Hermitian:

jj <- x %o% y
max(Mod(Im(jj+t(jj))))

then verify the Jordan identity:

LHS <- (x %o% y) %o% (x %o% x)
RHS <- x %o% (y %o% (x %o% x))
max(Mod(LHS-RHS))  # zero to numerical precision

showing that the Jordan identity is satisfied, up to a small numerical tolerance. However, $4\times 4$ octonionic matrices do not satisfy the Jordan identity:

x <- cprod(matrix(roct(16),4,4))
y <- cprod(matrix(roct(16),4,4)) # x and y are 4-by-4 Hermitian octonionic matrices
LHS <- (x %o% y) %o% (x %o% x)
RHS <- x %o% (y %o% (x %o% x))
max(Mod(LHS-RHS))  # miles off

Note on the print method

Above we have been using the default print method but it is possible to use a more compact notation, which is useful if a matrix has many short entries.

o <- rsomat()
o
options(show_onionmats_in_place = TRUE)
o

RobinHankin/onion documentation built on April 20, 2024, 2:05 p.m.