absdetA

absdetA denotes the absolute value of the determinant of a square matrix A, that is absdetA = |det(A)|. The determinant is defined for n by n matrices over a field (typically the real or complex numbers) and absdetA is its nonnegative magnitude. Geometrically, det(A) represents the oriented volume scaling factor of the linear transformation associated with A; absdetA is the same scaling factor when orientation is ignored.

If det(A) = 0, absdetA = 0 and A is singular; if det(A) ≠ 0, absdetA > 0 and A is invertible. Several algebraic properties carry over to absdetA: absdet(AB) = absdet(A) absdet(B) for square A and B of compatible size; absdet(cA) = |c|^n absdet(A) for an n×n matrix; absdet(A^T) = absdet(A); and absdet(A^{-1}) = 1/absdet(A) when A is invertible. Notably, det(A^T A) = det(A)^2 implies absdet(A) equals the product of A’s singular values, giving a robust measure of volume distortion.

In applications, absdetA often appears in change-of-variables formulas for integration, where the differential volume scales by the absolute value of a Jacobian determinant. It also features in numerical linear algebra and multivariate statistics as a scale factor in volume computations and in assessing invertibility and conditioning of transformations.

Example: for a 2×2 matrix A = [[a, b], [c, d]], det(A) = ad − bc and absdetA = |ad − bc|. This simple case illustrates the relation between determinant signs and their absolute magnitudes.