Determinants using Clifford algebra

In clifford: Arbitrary Dimensional Clifford Algebras

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

To cite the clifford package in publications please use @hankin2025_clifford_rmd. This short document shows how determinants can be calculated using Clifford algebra as implemented by the clifford R package; notation follows @hestenes1987. 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))
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))

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])

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]))

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))

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)

above we see numerical agreement.

References

clifford documentation built on July 5, 2026, 5:07 p.m.