Differentiating the pseudo determinant

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.