Canonical Forms of Matrices

Abstract

In Sect. 9.4, we saw that the transformation matrices are altered depending on the basis vectors we choose. Then a following question arises. Can we convert a (transformation ) matrix to as simple a form as possible by similarity transformation (s)? In Sect. 9.4, we have also shown that if we have two sets of basis vectors in a linear vector space $V^{n}$ we can always find a non-singular transformation matrix between the two. In conjunction with the transformation of the basis vectors , the matrix undergoes similarity transformation . It is our task in this chapter to find a simple form or a specific form (i.e., canonical form ) of a matrix as a result of the similarity transformation . For this purpose, we should first find eigenvalue (s) and corresponding eigenvector (s) of the matrix . Depending upon the nature of matrices, we get various canonical forms of matrices such as a triangle matrix and a diagonal matrix . Regarding any form of matrices, we can treat these matrices under a unified form called the Jordan canonical form .