Determinants using Clifford algebra

To cite the clifford package in publications please use Hankin (2022). This short document shows how determinants can be calculated using Clifford algebra as implemented by the clifford R package; notation follows Hestenes and Sobczyk (1987). The methods shown here are not computationally efficient compared with bespoke linear algebra implementations (such as used in base R).

Given a square matrix \(M\), we consider alternating forms on its column vectors. Requiring that the identity matrix map to \(+1\) gives a unique alternating form which we identify with the determinant of \(M\). Hestenes points out that wedge products of 1-vectors are alternating forms and further points out [equation 4.1, p33] that any alternating \(r\)-form \(\alpha_r=\alpha_r\left(a_1,a_2,\ldots,a_r\right)\) can be written in the form

\[\alpha_r\left(a_1,a_2,\ldots,a_r\right)= A_r^\dagger\cdot\left(a_1\wedge a_2\wedge\cdots\wedge a_r\right) \]

for some unique \(r\)-vector \(A_r^\dagger\) . We then define the determinant to be the \(r\)-form corresponding to \(A\) being the Clifford \(r\)-volume element. In the package this is easy to implement. Considering a \(3\times 3\) matrix as an example, we simply calculate the wedge product of its columns:

suppressMessages(library("clifford"))
set.seed(0)
(M <- matrix(rnorm(9),3,3))

##            [,1]       [,2]         [,3]
## [1,]  1.2629543  1.2724293 -0.928567035
## [2,] -0.3262334  0.4146414 -0.294720447
## [3,]  1.3297993 -1.5399500 -0.005767173

o <- as.1vector(M[,1]) ^ as.1vector(M[,2]) ^ as.1vector(M[,3])
Adag <- rev(e(seq_len(3)))
c(drop(Adag %.% o), det(M))

## [1] -1.031795 -1.031795

Above, we see numerical agreement [as a parenthetical note, we observe that the dagger is needed to get the correct sign if the dimension is odd]. Alternatively, we can examine the wedge product directly:

as.1vector(M[,1]) ^ as.1vector(M[,2]) ^ as.1vector(M[,3])

## Element of a Clifford algebra, equal to
## - 1.031795e_123

Note that the wedge product is given as a Clifford volume element. We can extract its coefficient (which would be the determinant of M) using coeffs():

coeffs(as.1vector(M[,1]) ^ as.1vector(M[,2]) ^ as.1vector(M[,3]))

## [1] -1.031795

Just as a consistency check, we evaluate the wedge product of a set of basis vectors, effectively calculating the determinant of \(I_3\):

coeffs(e(1) ^ e(2) ^ e(3))

## [1] 1

We see \(+1\), as expected. It is possible to consider larger matrices:

cliff_det <- function(M){
  o <- as.1vector(M[,1])
  for(i in 2:nrow(M)){
    o <- o ^ as.1vector(M[,i])
  }
  return(coeffs(o))
}

Then

M <- matrix(rnorm(100),10,10)
LHS <- det(M)
RHS <- cliff_det(M)
c(LHS,RHS,LHS-RHS)

## [1] 1.478400e+02 1.478400e+02 1.136868e-13

above we see numerical agreement.

References

Hankin, R. K. S. 2022. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.

Hestenes, D., and G. Sobczyk. 1987. Clifford Algebra to Geometric Calculus. Kluwer.

Robin K. S. Hankin

References