Jordan Canonical Form, Generalised Eigen Vectors and its Applications

Abstract

Introduction

X. ACKNOWLEDGEMENT

The project entitled ‘Jordan Canonical Form, Generalised Eigenvectors and its Applications’ comprises of my research work. Firstly, I would like to express my sincere gratitude towards my supervisor, Professor Diptiman Saha, Associate Professor, Department of Mathematics, St. Xavier’s College (Autonomous) Kolkata, without whom this project would not have been possible. His uninhibited availability, constant guidance and encouragement has helped me to pursue this topic diligently and without any difficulty.

I would also like to thank Professor Sucharita Roy, H.O.D, Assistant Professor, Department of Mathematics, St. Xavier’s College (Autonomous)Kolkata, for providing me with the opportunity to undertake this topic as my project.

I owe my deepest gratitude towards my parents for being supportive throughout the process of this project.

Lastly, my heartfelt thanks to my friends for their constant support and cooperation. Their timely help and friendship shall always be remembered.

Conclusion

The Jordan Canonical Form describes the structure of an arbitrary linear transformation on a finite- dimensional vector space over an algebraically closed field. Here we develop it only using the basic concepts of linear algebra, with no reference to determinants or ideals of polynomials. In this project, we have talked about how to explicitly compute Jordan forms. A. Uses of JCF 1) Over an algebraically closed field, which matrices are diagonalisable? Diagonalisable matrices are those matrices which have all regular eigenvalues i.e. geometric multiplicity of each eigen value is equal to its algebraic multiplicity. But there exist non-diagonalisable matrices too. For such non-diagonalisable matrices, there exist atleast one eigenvalue for which geometric multiplicity is less than its algebraic multiplicity. Diagonalisable matrices are similar to a diagonal matrix and non-diagonalisable matrices are similar to JCF of the matrix. Diagonal matrix is special form of JCF where each Jordan block is of size 1. The question posed above can also be answered by looking at the structure of JCF. JCF of a diagonalisable matrix has all Jordan blocks of size 1. JCF of a non-diagonalisable matrix has atleast one Jordan block of size ?2. 2) The JCF presents all the important data about a matrix-the list of eigenvalues, eigendimension , generalised eigendimension associated to each eigen value and the minimal and characteristic polynomials in a readable form. 3) When does minimal polynomial coincide with the characteristic polynomial? The characteristic polynomial of a matrix Anxn equals the minimal polynomial of Anxn if and only if the dimension of each eigenspace of A is 1 i.e. the matrix has n distinct eigenvalues. If a matrix has n distinct eigenvalues, then JCF of the matrix will have all Jordan blocks of size 1 corresponding to n distinct eigenvalues. Therefore, also by looking at the structure of JCF of a matrix, we can say whether the minimal polynomial coincide with the characteristic polynomial or not? 4) Over an algebraically closed field F, the JCF is a complete invariant for conjugacy. This means the following for A, B ? M (n, F), we have that A is conjugate to B iff the JCF of A and B are the same upto the permutation of the blocks. The fact that JCF is a complete invariant for conjugacy is all the more interesting since the minimal and the characteristic polynomial together do not form a complete conjugacy invariant. Also, the JCF over C is a complete conjugacy invariant for square matrices over R even though R is not algebraically closed. 5) JCF is useful in solving the system of linear differential equations ?????=Ax, where A need not be diagonalisable.

References

[1] Linear Algebra - A. Ramachandra Rao and P. Bhimsankara [2] Linear Algebra - Hoffman and Kunze [3] Linear Algebra-Arnold J. Insel , Lawrence E. Spence and Stephen H. Friedberg [4] Linear Algebra and its Applications -David C lay and Steven R Lay [5] Differential Equations and Dynamical systems by Lawrence Perko

Copyright

Copyright © 2023 Bratasee Bhunia, Prof. Diptiman Saha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.