Procedures: Weak Drazin inverse of a square matrix
(%i1) WeakDrazinInverse ( A ) : = block ( [ n , m , d , s , P , X ] ,
   if not ( matrixp ( A ) ) then error ( "Matrix expected" ) ,
   [ n , m ] : matrix_size ( A ) ,
   if n # m then error ( "Square matrix expected" ) ,
   X : ident ( n ) − s · A ,
   P : expand ( determinant ( X ) ) ,
   d : hipow ( P , s ) ,
   coeff ( expand ( adjoint ( X ) ) , s , d − 1 ) / coeff ( P , s , d )
) $
Example
(%i5) print ( "Matrix:" ) $
M : matrix ( [ m11 , m12 , m13 ] , [ m21 , 0 , 0 ] , [ m31 , 0 , 0 ] ) ;
print ( "Weak Drazin inverse:" ) $
WeakDrazinInverse ( M ) ;

\[\]\[\mbox{Matrix}:\]

\[\operatorname{(M) }\begin{pmatrix}\ensuremath{\mathrm{m11}} & \ensuremath{\mathrm{m12}} & \ensuremath{\mathrm{m13}}\\ \ensuremath{\mathrm{m21}} & 0 & 0\\ \ensuremath{\mathrm{m31}} & 0 & 0\end{pmatrix}\]\[\mbox{Weak Drazin inverse:}\]

\[\operatorname{ }\begin{pmatrix}0 & \frac{\ensuremath{\mathrm{m12}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & \frac{\ensuremath{\mathrm{m13}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}}\\ \frac{\ensuremath{\mathrm{m21}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & -\frac{\ensuremath{\mathrm{m11}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & 0\\ \frac{\ensuremath{\mathrm{m31}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & 0 & -\frac{\ensuremath{\mathrm{m11}}}{-\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}-\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}}\end{pmatrix}\]

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