detInv1

In mathematics and numerical analysis, detInv1 denotes the determinant of the inverse of a square invertible matrix A, commonly written as det(A^{-1}). This quantity provides a scalar measure of how volumes are scaled by the inverse linear transformation associated with A.

A fundamental property is det(A^{-1}) = 1/det(A) for any invertible matrix A. Consequently, detInv1 is defined exactly when det(A) ≠ 0. If A has eigenvalues λ1, λ2, ..., λn, then the eigenvalues of A^{-1} are 1/λi, and detInv1 = ∏(1/λi) = 1/det(A). This links detInv1 directly to the determinant of A and to the spectral characteristics of the matrix.

Computationally, detInv1 can be obtained by either directly computing det(A) and taking its reciprocal, i.e., detInv1 = 1/det(A), or by performing a factorization such as LU or Cholesky and combining the factors to form det(A) and then inverting it. In practice, computing detInv1 via det(A) is typically preferred, since det(A^{-1}) does not provide separate numerical advantages and can amplify errors when det(A) is close to zero.

Numerical considerations include potential instability when det(A) is very small or very large, which makes detInv1 potentially large or ill-conditioned. In such cases, using logarithms of determinants (log det) or the log-determinant approach can improve numerical stability and avoid overflow or underflow.

Applications of detInv1 arise in areas such as change-of-variables in integration, volume transformations under linear maps, and conditioning analyses of linear systems, where understanding the scale induced by the inverse transformation is relevant. See also det, inv, and log-determinant.