detEuclidska

detEuclidska is a term used in some mathematical contexts to denote the determinant of a square matrix, interpreted as the oriented scaling factor of a linear transformation on Euclidean space. It is defined for matrices over a field, typically the real or complex numbers.

Definition and properties: For an n×n matrix A with columns a1, …, an in R^n, detEuclidska(A) equals the signed n-dimensional volume of the parallelepiped spanned by these vectors. It is the unique function that is multilinear in the columns, alternating, and normalized so that detEuclidska(I) = 1. If the columns are linearly dependent, detEuclidska(A) = 0. For any invertible P, detEuclidska(P⁻¹AP) = detEuclidska(A). These properties imply that detEuclidska(A) = 0 precisely when A is not invertible, and that detEuclidska(A) equals the product of A’s eigenvalues (counted with multiplicity) when over the complex field.

Computation: Determinants can be computed by expansion by minors, or more practically via LU decomposition, where detEuclidska(A) equals the product of the diagonal entries of U, adjusted by the sign change from any row swaps. Row operations provide intuition: swapping rows changes the sign, scaling a row scales the determinant, and adding a multiple of one row to another leaves it unchanged.

Geometric interpretation and applications: The absolute value |detEuclidska(A)| equals the volume (area in 2D) of the parallelepiped spanned by the column vectors, giving a measure of the linear transformation’s effect on volume. Consequently, detEuclidska serves to test invertibility, perform change-of-basis computations, and relate linear transformations to geometric volume.

See also: determinant, linear algebra, Euclidean space, volume, change of basis.