Author: Robin K. S. Hankin

# Determinants using Clifford algebra

#### Robin K. S. Hankin

To cite the `clifford` package in publications please use Hankin (2025). 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. 2025. “Clifford Algebra in R: Introducing the clifford Package.” Advances in Applied Clifford Algebra 35 (51). https://doi.org/https://doi.org/10.1007/s00006-025-01403-9.
Hestenes, D., and G. Sobczyk. 1987. Clifford Algebra to Geometric Calculus. Kluwer.