Differentiating the pseudo determinant

TL;DR

This paper introduces a class of derivatives for the pseudo determinant of Hermitian matrices, establishing a canonical derivative that generalizes the classic determinant gradient to rank-deficient matrices.

Contribution

It defines a non-empty class of derivatives for the pseudo determinant and identifies a unique canonical member, extending determinant calculus to singular matrices.

Findings

The canonical derivative is given by Det(A)A^+.

The class of derivatives is non-empty and well-defined.

Applications include maximum likelihood estimation for degenerate Gaussian distributions.

Abstract

A class of derivatives is defined for the pseudo determinant $Det(A)$ of a Hermitian matrix $A$. This class is shown to be non-empty and to have a unique, canonical member $\mathbf{\nabla Det}(A)=Det(A)A^+$, where $A^+$ is the Moore-Penrose pseudo inverse. The classic identity for the gradient of the determinant is thus reproduced. Examples are provided, including the maximum likelihood problem for the rank-deficient covariance matrix of the degenerate multivariate Gaussian distribution.