Weak Drazin Inverse

In [1], Campbell and Meyer defined some modifications of the classical Drazin inverse. We describe the calculation of one particular simple generalized inverse of this type and call this matrix a weak Drazin inverse. More details can be found in [2, Section 3.1] (in german). This weak Drazin inverse can be used to contruct a generalized transfer function for singular descriptor systems (i.e., for $\det(sE-A)\equiv0$), see [3].

Example

Consider the $3\times3$ matrix $M=\left( \begin{array}{ccc} m_{11}&m_{12}&m_{13}\\ m_{21}&0 &0\\ m_{31}&0 &0 \end{array} \right)$ over a field $\mathbb{K}$. From the determinant $\det(I-\mu M)= 1 - {m_{11}}\,\mu-{m_{12} m_{21}}\,{\mu^2}-{m_{13} m_{31}}\,{\mu^2}$ we obtain the coefficient $p_2:=-{m_{12} m_{21}}-{m_{13} m_{31}}$. Assume that $p_2\neq0$. From this assumption we get $d=2$ und $\mbox{ind}(M)=1$. The adjoint matrix (i.e., the transpose of the matrix of cofactors) of $(I-\mu M)$ reads
$\mbox{adj}(I-\mu M)= \left( \matrix{ 1 & { m_{12}}\,\mu & { m_{13}}\,\mu \cr { m_{21}}\,\mu & 1 - { m_{11}}\,\mu - { m_{13}}\,{ m_{31}}\,{\mu^2} & { m_{13}}\,{ m_{21}}\,{\mu^2} \cr { m_{31}}\,\mu & { m_{12}}\,{ m_{31}}\,{\mu^2} & 1 - { m_{11}}\,\mu - { m_{12}}\,{ m_{21}}\,{\mu^2} \cr } \right)$
Selecting the numerator of the coefficient matrix $R_1$ (i.e., potency $\mu^1$), we obtain the weak Drazin inverse $M^d$: $M^d=\frac{1}{{m_{12} m_{21}}+{m_{13} m_{31}}} \left( \matrix{ 0 & { m_{12}} & { m_{13}} \cr { m_{21}} & -{ m_{11}} & 0 \cr { m_{31}} & 0 & -{ m_{11}} \cr } \right)$

References

1. Campbell, S. L. & Meyer, C. D.: Weak Drazin inverses. Linear Algebra and its Applications, 1978, 20, 167–178.
2. Röbenack, K.: Beitrag zur Analyse von Deskriptorsystemen. Shaker-Verlag, 1999 (ISBN: 978-3-8265-6795-7) mehr (in german).
3. Reinschke, K. & Röbenack, K.: Graphentheoretische Analyse des Eingangs-Ausgangsverhaltens singulärer Deskriptorsysteme.
Tagungsband 42. Internationales Wissenschaftliches Kolloquium, Band 3, TU Ilmenau, 1997, 107-112 (in german).