sqrtdet

Sqrtdet is shorthand for the square root of the determinant of a square matrix, commonly written as sqrt(det(A)) for a matrix A. When A is symmetric positive semidefinite (or positive definite), det(A) is nonnegative and sqrt(det(A)) is a real, nonnegative quantity. In terms of eigenvalues, if the eigenvalues λ1, λ2, ..., λn of A are all nonnegative, then sqrt(det(A)) = ∏i sqrt(λi). Equivalently, if A = L L^T is a Cholesky factorization with L lower triangular, then sqrt(det(A)) equals the product of the diagonal elements of L, i.e., sqrt(det(A)) = ∏i Lii.

Computationally, directly computing det(A) can be numerically unstable or expensive. A common approach is to use a Cholesky decomposition A = L L^T for symmetric positive definite A, in which case det(A) = (∏i Lii)^2 and sqrt(det(A)) = ∏i Lii. Alternatively, one may compute the log-determinant log det(A) and then take its half to obtain log(sqrtdet) = 0.5 log det(A). This avoids overflow or underflow and is standard in probabilistic modeling and numerical linear algebra. If A is singular or near-singular, det(A) may be zero (or numerically near zero), giving sqrtdet ≈ 0; numerical regularization or rank checks may be required.

Applications of sqrtdet appear in probability densities and statistical models that involve normalizing constants. For example, in multivariate normal distributions, the normalization term includes sqrt((2π)^n det(Σ)). In Gaussian processes and determinantal point processes, determinants and their square roots influence likelihoods, priors, and normalization factors. Care is needed to ensure the matrix is appropriate for taking a square root of its determinant, typically requiring symmetry and positive definiteness or appropriate regularization.