The Drazin inverse as a gradient (Q2266753)

If A is an invertible \(n\times n\) matrix, X an \(n\times n\) matrix variable, and \(| X| =\det X\), then \(\ell n | X|\) is differentiable near A and \(\nabla_ x\) \(\ell n | X|\) at \(X=A\) is \((A^{-1})^ T\). This paper examines various analogues and related questions for the Drazin inverse \(A^ d\). In particular, a matrix function \(W_ A(X)\), dependent on A, is derived so that \((A^ d)^ T=\nabla_ x\ell n W_ A(X)\) at \(X=A\). The matrix function \(W_ A(X)\) is not continuous in A. Also \(\nabla_ x\) may need to be interpreted formally over some fields.