logdeterminant

Log-determinant, or logdet, of a square matrix A is the natural logarithm of its determinant, written as log det(A). It is real-valued when det(A) > 0, and is most commonly defined for symmetric positive definite matrices, for which det(A) > 0. If A is singular, det(A) = 0 and log det is undefined in the real sense; in some contexts a pseudodeterminant or a complex-valued extension may be used.

Equivalences: if A has eigenvalues λ_i, then det(A) = ∏ λ_i and log det(A) = ∑ log λ_i. If A is positive definite, log det(A) also equals tr(log A), where log A is the matrix logarithm. These forms connect det and the eigenstructure or the matrix logarithm.

Computation: for SPD A, a stable method is Cholesky decomposition A = L L^T and log det(A) = 2 ∑_i log L_ii. In general invertible A, log det A can be obtained from LU decomposition as log det A = ∑ log|U_ii| + log|P|, where P is a permutation matrix. In numerical practice, direct det computation is avoided in favor of these decompositions for stability.

Differentiation and gradients: the derivative with respect to A is ∂ log det A / ∂ A = (A^{-T}); for symmetric A this reduces to A^{-1}. This makes log det A a smooth, differentiable function on the space of invertible matrices, with a gradient aligned with the inverse of A.

Applications: the log-determinant appears in Gaussian log-likelihoods, Bayesian statistics, Gaussian processes, and various optimization problems, providing a scale-invariant measure of volume and improving numerical stability in high-dimensional settings.

See also: determinant, matrix logarithm, Cholesky decomposition, LU decomposition.