TITLE:
The Study of Root Subspace Decomposition between Characteristic Polynomials and Minimum Polynomial
AUTHORS:
Lilong Kang, Yu Wang, Yingling Liu
KEYWORDS:
Characteristic Polynomial, Minimum Polynomial, Root Subspace
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.14 No.7,
July
8,
2024
ABSTRACT: Let
A
be the linear transformation on the linear space V in the field P,
V
λ
i
be the root subspace corresponding to the characteristic polynomial of the eigenvalue
λ
i
, and
W
λ
i
be the root subspace corresponding to the minimum polynomial of
λ
i
. Consider the problem of whether
V
λ
i
and
W
λ
i
are equal under the condition that the characteristic polynomial of
A
has the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that
V
λ
i
=
W
λ
i
. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”