A Directed Graph and MATLAB Generation of the Jordan Canonical Form for a Class of Zero-One Matrices

The numerically challenging problem of generating the Jordan canonical form is attacked exclusively for a class of zero-one square matrices with the property that each column has at most one nonzero element. The approach is based on the generation and analysis of the directed graph having the matrix in question as its adjacency matrix. The computational investigation is carried out by developing a MATLAB experimental toolbox. Fortunately, it has been found that this toolbox strikingly outperformed the built-in MATLAB function “jordan” since the former successfully executed for square matrices of order up to 1000 while the latter lingered for matrices of order 60. Since the Collatz matrix generated by the Collatz function is a special case of the zero-one investigated matrix, the obtained results are applicable to this important matrix which appears in some problems in signal and image processing. The open problems related to Collatz conjecture are computationally addressed and some empirical results emerged. An empirical formula is devised for the number of connected components in the Collatz graph.