Column-wise unit normalisation of a matrix

Rescales each column of V to unit Euclidean norm. Columns whose Euclidean norm falls below \(10^{-15}\) (effectively the zero vector) trigger an error rather than silently producing NaN.

Usage

normalize_vectors(V)

Arguments

V

A numeric matrix.

Value

A numeric matrix of the same dimensions as V, with each column rescaled so that \(\|V[, j]\|_2 = 1\).

Details

Used as a preprocessing step throughout the package. The geometry of a simplicial cone is invariant under positive rescaling of its generators, so unit-normalising before the dispatch removes a degree of freedom that would otherwise leak into the diagnostic numbers (e.g. condition number of the Gram matrix).

References

Strang, G. (2016). Introduction to Linear Algebra, 5th edition. Wellesley-Cambridge Press. ISBN 978-0980232776.

See also

compute_associated_matrix and compute_dot_product_matrix for downstream consumers; compute_solid_angle for the dispatcher that calls this helper as part of its preprocessing.

Examples

V <- cbind(c(3, 4), c(1, 1))
V_unit <- normalize_vectors(V)
apply(V_unit, 2, function(v) sqrt(sum(v^2)))   # both equal 1
#> [1] 1 1