Generalized Cross Product via Determinants

Computes a vector orthogonal to all rows of a matrix using a determinant-based construction. This generalizes the cross product to higher dimensions.

Usage

crossProdN(A)

Arguments

A: a numeric or complex vector of length 2, or a matrix of dimension $n x (n+1)$.

Value

a numeric or complex vector of length $n+1$.

Details

For a matrix $A$ with dimensions $n x (n+1)$, the result is a vector in $R^{n+1}$ orthogonal to all rows of $A$. The components are given by: $$ v_i = (-1)^{i+1} \det(A_{-i}) $$ where $A_{-i}$ is the matrix obtained by removing the $i$-th column.

For a vector of length 2, the function returns the perpendicular vector $(a_2, -a_1)$, consistent with the formula above.

Internally, the function computes a nullspace vector using SVD (which is numerically stable also for ill-conditioned input) and rescales it to match the magnitude of the determinant-based generalized cross product. For numeric input, the sign is chosen to reproduce the orientation of the determinant formula (and hence anticommutativity: swapping two rows of $A$ flips the sign of the result). For complex input, where det() is not available, the sign is fixed by the convention that the first component with non-zero modulus has a positive real part.

Examples

# 2D case
crossProdN(c(1, 2))
#> [1]  2 -1

# 3D case (standard cross product)
A <- matrix(c(1,0,0,
              0,1,0), nrow = 2, byrow = TRUE)
crossProdN(A)
#> [1] 0 0 1

# swapping rows flips the sign (anticommutativity)
crossProdN(A[2:1, ])
#> [1]  0  0 -1