Onionic matrices

In onion: Octonions and Quaternions

knitr::opts_chunk$set(echo = TRUE)
library("onion") 

Onionic matrices

The onion package allows one to define and manipulate matrices whose elements are quaternions or octonions. Function romat() 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, a 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

Consider the following $3\times 3$ octonionic matrices:

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))

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.

onion documentation built on Feb. 11, 2021, 9:06 a.m.