Using Gauss-Jordan elimination to compute the index,generalized nullspaces,and Drazin inverse

We develop and analyze a new algorithm that computes bases for the null spaces of all powers of a given matrix, as well as its index. The algorithm uses row operations and “shuffling” steps in which rows of pairs of matrices are interchanged. In particular, the new algorithm may be viewed as an extension of the classic Gauss-Jordan elimination method for inverting a nonsingular matrix. It is also shown that the Drazin inverse has a simple representation in terms of the output of the algorithm and the original matrix.