Drazin Inverse

On this page we shortly discuss definition and calculation of the Drazin inverse.


Let $M\in\mathbb{K}^{n\times n}$ be a square matrix over the field $\mathbb{K}$ with $k=\mbox{ind}(M)$. A matrix $D\in\mathbb{K}^{n\times n}$ is called Drazin inverse of $M$, if

\[\begin{array}{lcl} MD &=& DM\\ DMD &=& D\\ DM^{k+1} &=& M^{k}. \end{array}\]

Notation: $M^{D}$.

If $\mathbb{K}$ is a hausdorff topological field, the Drazin inverse can be computed by

\[M^{D}=\lim_{\lambda\to0}\left(\lambda I+M^{p+1}\right)^{-1}M^{p}\quad\mbox{with}\quad p\geq k,\]

see [1].

Example 1

The $3\times3$ matrix

\[M=\left(\begin{array}{ccc} m_{11} & m_{12} & m_{13}\\ m_{21} & 0 & 0\\ m_{31} & 0 & 0 \end{array}\right)\]

has the index $k=\mbox{ind}(M)=1$. The Drazin inverse is given below and in [2, p. 69].

Example 2

For the $4\times4$ matrix

\[M= \pmatrix{0&0&0&{\it m_{14}}\cr 0&0&{\it m_{23}}&{\it m_{24}}\cr {\it m_{31}}&{\it m_{32}}&0&{\it m_{34}}\cr 0&0&0&0\cr }\]

we have $\mbox{rank}(M)=3$ and $\mbox{rank}(M^2)=\mbox{rank}(M^3)=2$. Therefore, the matrix has index $k=\mbox{ind}(M)=2$.


  1. C. D. Meyer: Limits and the index of a square matrix.
    SIAM J. Appl. Math. 26, 1974, pp. 199-216.
  2. Röbenack, K.: Beitrag zur Analyse von Deskriptorsystemen.
    Shaker-Verlag, 1999, ISBN: 978-3-8265-6795-7, (in german).
  3. Röbenack, K.; Reinschke, K. On generalized inverses of singular matrix pencils.
    Applied Mathematics and Computer Science, 2011, 21(1), pp. 161-172