International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 10 Issue: 07 | July 2023
p-ISSN: 2395-0072
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A MATLAB Computational Investigation of the Jordan Canonical Form of a Class of Zero-One Matrices Raouth R. Ghabbour1, Magdy Tawfik Hanna2 1 Demonstrator, Department of Engineering Mathematics and Physics, Fayoum University, Fayoum, Egypt, 2 Professor, Department of Engineering Mathematics and Physics, Fayoum University, Fayoum, Egypt,
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Abstract – The difficult task of creating the Jordan
problems in linear algebra are used. The use of combinatorial and graph theoretic methods for understanding the Jordan canonical form has a long history. In 1837, Jacobi showed that a square matrix of order n is similar to an upper triangular matrix. Many proofs of the Jordan form rely on this result. Directed graphs may help in dealing with several problems in neural networks and learning systems, network science and engineering, and automatic control[1]–[8]. Previously, Cardon and Tuckfield [9] proposed an algorithm for finding the Jordan canonical form of a class of non-diagonalizable zero-one square matrices - with the property that each column has at most one nonzero element - using the directed graph (digraph) tool. The Jordan form was obtained for the adjacency matrix describing the directed graph.
canonical form is handled only for a class of zero-one square matrices with the additional property that each column has at most one nonzero element. The method is based on the construction and analysis of the adjacency matrix of directed graph. The computational study focuses primarily on the creation of the Jordan canonical form and modal matrix which contains eigenvectors and generalized eigenvectors. The computation's accuracy is measured by calculating the difference between the provided matrix and that created by combining the Jordan form and the modal matrix. Both the maximum element in absolute value and the Frobenius norm of the difference matrix - defined as the difference between the given and computed matrices - are used as error measures. The computational investigation is carried out by creating a set of MATLAB functions that can be seen as a novel contributed toolbox. Fortunately, it was discovered that this toolbox outperformed the built-in MATLAB function "jordan" because the contributed toolbox successfully processed square matrices of order up to 1000 while the built-in function of the MATLAB lingered for matrices of order between 40 to 60.
It is possible to date the beginning of graph theory to 1735, when the Swiss mathematician Leonhard Euler found an answer to the Königsberg bridge puzzle [10], [11]. The Königsberg bridge problem was an old puzzle that involved trying to find a way over each of the seven bridges that span a branched river that flows by an island without using any of them more than once. As shown in Figure 1, there were seven bridges in the city crossing a waterway between two banks and two islands. The question was whether it was practical to cross every bridge just once and return to the starting point of the journey. The response was no, it is impossible to cross every bridge precisely once. It should be mentioned that the Königsberg bridge problem is not a simple graph example because there are multiple edges connecting the same two nodes, which is acceptable in a multigraph. In a multigraph with four vertices and seven edges, the issue mathematically comes down to finding an Eulerian cycle [11].
Key Words: Directed graphs, Jordan canonical form, zero-one matrices.
1.INTRODUCTION Jordan canonical form is a specific type of upper triangular matrix's representation of a linear transformation over a finite dimensional vector space. Every such linear transformation has a distinct Jordan canonical form with the nice properties. It is simple to explain and well-suited for calculation. Every square matrix is similar to a unique matrix in Jordan canonical form, because similar matrices correspond to representations of the same linear transformation with respect to different bases, according to the change of basis theorem. Jordan canonical form is a generalisation of diagonalizability to arbitrary linear transformations (or matrices); in fact, the Jordan canonical form of a diagonalizable linear transformation (or matrix) is a diagonal matrix also. To compute Jordan canonical form, most researchers in matrix theory and most users of its methods are aware of the importance of graphs in linear algebra. This can be seen from the large number of papers in which graph theoretic methods for solving
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