Chapter 12
Canonical Forms of Matrices
In Sect. 11.4 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. 11.4 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.
12.1 Eigenvalues and Eigenvectors
An eigenvalue problem is one of important subjects of the theory of linear vector
spaces. Let A be a linear transformation on V
n . The resulting matrix gets to several
different kinds of canonical forms of matrices. A typical example is a diagonal
matrix. To reach a satisfactory answer, we start with so-called an eigenvalue
problem.
Suppose that after the transformation of x we have
A x
ð Þ ¼ αx,
ð12:1Þ
where α is a certain (complex) number. Then we say that α is an eigenvalue and that
x is an eigenvector that corresponds to the eigenvalue α. Using a notation of (11.37)
of Sect. 11.2, we have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_12
459
Canonical Forms of Matrices
In Sect. 11.4 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. 11.4 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.
12.1 Eigenvalues and Eigenvectors
An eigenvalue problem is one of important subjects of the theory of linear vector
spaces. Let A be a linear transformation on V
n . The resulting matrix gets to several
different kinds of canonical forms of matrices. A typical example is a diagonal
matrix. To reach a satisfactory answer, we start with so-called an eigenvalue
problem.
Suppose that after the transformation of x we have
A x
ð Þ ¼ αx,
ð12:1Þ
where α is a certain (complex) number. Then we say that α is an eigenvalue and that
x is an eigenvector that corresponds to the eigenvalue α. Using a notation of (11.37)
of Sect. 11.2, we have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_12
459