Procedures: Index and Drazin inverse of a square matrix
(%i1) MatrixIndex ( A ) : = block ( [ n , m , i , r0 , r1 , B ] ,
   if not ( matrixp ( A ) ) then error ( "Matrix expected" ) ,
   [ n , m ] : matrix_size ( A ) ,
   if n # m then error ( "Square matrix expected" ) ,
   r0 : n ,
   B : ident ( n ) ,
   for i : 0 thru n do (
       B : B . A ,
       r1 : rank ( B ) ,
       if is ( r0 = r1 ) then return ( i ) ,
       r0 : r1
   )
) $
(%i2) DrazinInverse ( A ) : = block ( [ n , k , s ] ,
   k : MatrixIndex ( A ) ,
   n : length ( A ) ,
   limit ( invert ( s · ident ( n ) + A ^ ^ ( k + 1 ) ) . ( A ^ ^ k ) , s , 0 )
) $
Example 1
(%i5) M1 : matrix ( [ m11 , m12 , m13 ] , [ m21 , 0 , 0 ] , [ m31 , 0 , 0 ] ) ;
print ( "Index: " , MatrixIndex ( M1 ) ) $
DrazinInverse ( M1 ) ;

\[\operatorname{(M1) }\begin{pmatrix}\ensuremath{\mathrm{m11}} & \ensuremath{\mathrm{m12}} & \ensuremath{\mathrm{m13}}\\ \ensuremath{\mathrm{m21}} & 0 & 0\\ \ensuremath{\mathrm{m31}} & 0 & 0\end{pmatrix}\]\[\mbox{Index: 1}\]

\[\operatorname{ }\begin{pmatrix}0 & \frac{\ensuremath{\mathrm{m12}}}{\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}+\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & \frac{\ensuremath{\mathrm{m13}}}{\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}+\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}}\\ \frac{\ensuremath{\mathrm{m21}}}{\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}+\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & -\frac{\ensuremath{\mathrm{m11}}\, \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}}{{{\ensuremath{\mathrm{m13}}}^{2}} {{\ensuremath{\mathrm{m31}}}^{2}}+2 \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m21}}\, \ensuremath{\mathrm{m31}}+{{\ensuremath{\mathrm{m12}}}^{2}} {{\ensuremath{\mathrm{m21}}}^{2}}} & -\frac{\ensuremath{\mathrm{m11}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m21}}}{{{\ensuremath{\mathrm{m13}}}^{2}} {{\ensuremath{\mathrm{m31}}}^{2}}+2 \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m21}}\, \ensuremath{\mathrm{m31}}+{{\ensuremath{\mathrm{m12}}}^{2}} {{\ensuremath{\mathrm{m21}}}^{2}}}\\ \frac{\ensuremath{\mathrm{m31}}}{\ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}+\ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m21}}} & -\frac{\ensuremath{\mathrm{m11}}\, \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m31}}}{{{\ensuremath{\mathrm{m13}}}^{2}} {{\ensuremath{\mathrm{m31}}}^{2}}+2 \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m21}}\, \ensuremath{\mathrm{m31}}+{{\ensuremath{\mathrm{m12}}}^{2}} {{\ensuremath{\mathrm{m21}}}^{2}}} & -\frac{\ensuremath{\mathrm{m11}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m31}}}{{{\ensuremath{\mathrm{m13}}}^{2}} {{\ensuremath{\mathrm{m31}}}^{2}}+2 \ensuremath{\mathrm{m12}}\, \ensuremath{\mathrm{m13}}\, \ensuremath{\mathrm{m21}}\, \ensuremath{\mathrm{m31}}+{{\ensuremath{\mathrm{m12}}}^{2}} {{\ensuremath{\mathrm{m21}}}^{2}}}\end{pmatrix}\]

Example 2
(%i8) M2 : matrix ( [ 0 , 0 , 0 , m14 ] , [ 0 , 0 , m23 , m24 ] , [ m31 , m32 , 0 , m34 ] , [ 0 , 0 , 0 , 0 ] ) ;
print ( "Index: " , MatrixIndex ( M2 ) ) $
DrazinInverse ( M2 ) ;

\[\operatorname{(M2) }\begin{pmatrix}0 & 0 & 0 & \ensuremath{\mathrm{m14}}\\ 0 & 0 & \ensuremath{\mathrm{m23}} & \ensuremath{\mathrm{m24}}\\ \ensuremath{\mathrm{m31}} & \ensuremath{\mathrm{m32}} & 0 & \ensuremath{\mathrm{m34}}\\ 0 & 0 & 0 & 0\end{pmatrix}\]\[\mbox{Index: 2}\]

\[\operatorname{ }\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & \frac{1}{\ensuremath{\mathrm{m32}}} & \frac{\ensuremath{\mathrm{m24}}\, \ensuremath{\mathrm{m32}}+\ensuremath{\mathrm{m14}}\, \ensuremath{\mathrm{m31}}}{\ensuremath{\mathrm{m23}} {{\ensuremath{\mathrm{m32}}}^{2}}}\\ \frac{\ensuremath{\mathrm{m31}}}{\ensuremath{\mathrm{m23}}\, \ensuremath{\mathrm{m32}}} & \frac{1}{\ensuremath{\mathrm{m23}}} & 0 & \frac{\ensuremath{\mathrm{m34}}}{\ensuremath{\mathrm{m23}}\, \ensuremath{\mathrm{m32}}}\\ 0 & 0 & 0 & 0\end{pmatrix}\]

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