On the Derivative of the Drazin Inverse of a Complex Matrix
Robert E. Hartwig, Jim Shoaf · SIAM Journal on Mathematical Analysis · 1979
It is shown that the derivative of the Drazin inverse of a differentiable matrix $A(t)$ exists for all values of t in the domain of definition except for the kernels of the nontrivial eigenvalues. Expressions are found for this derivative in terms of the characteristic polynomial, the spectral components and the matrices A, $A^0 $ and $A^d $ respectively. A short proof is given for Stewart’s theorem on the continuity of the Moore–Penrose inverse, and a formula corresponding to Wedin’s formula is given for the Drazin inverse.