Drazin Inverse of a square Matrix

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

Definition

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 obtain the following Drazin inverse [2, p. 69]: \[M^D=\left( \matrix{ 0 & {{{ m_{12}}}\over {{ m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}}}} & {{{ m_{13}}}\over {{ m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}}}} \cr {{{ m_{21}}}\over {{ m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}}}} & -{{{ m_{11}}\,{ m_{12}}\,{ m_{21}}}\over {{{\left( { m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}} \right) }^2}}} & -{{{ m_{11}}\,{ m_{13}}\,{ m_{21}}}\over {{{\left( { m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}} \right) }^2}}} \cr {{{ m_{31}}}\over {{ m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}}}} & -{{{ m_{11}}\,{ m_{12}}\,{ m_{31}}}\over {{{\left( { m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}} \right) }^2}}} & -{{{ m_{11}}\,{ m_{13}}\,{ m_{31}}}\over {{{\left( { m_{12}}\,{ m_{21}} + { m_{13}}\,{ m_{31}} \right) }^2}}} \cr } .\right)\]

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 $k=\mbox{ind}(M)=2$. The Drazin inverse is given by \[ M^D= \pmatrix{0&0&0&0\cr 0&0&{{1}\over{{\it m_{32}}}}&{{{\it m_{24}}\, {\it m_{32}}+{\it m_{14}}\,{\it m_{31}}}\over{{\it m_{23}}\, {\it m_{32}}^2}}\cr {{{\it m_{31}}}\over{{\it m_{23}}\,{\it m_{32}} }}&{{1}\over{{\it m_{23}}}}&0&{{{\it m_{34}}}\over{{\it m_{23}}\, {\it m_{32}}}}\cr 0&0&0&0\cr } \]

References

  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) mehr.
  3. Röbenack, K.; Reinschke, K. On generalized inverses of singular matrix pencils. Applied Mathematics and Computer Science, 2011, 21(1), S. 161-172

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