Study on Some Properties of Anti-centrosymmetric Matrices
Abstract
In this paper, the anti-centrosymmetric matrices have been researched. According to the structural characteristics of the anti-centrosymmetric matrix, some new methods have been used to prove the necessary and sufficient conditions of a matrix being anti-centrosymmetric and its properties of eigenvalue and eigenvector; the nonsingularity of the anti-centrosymmetric matrices have been discussed,that the odd order anti-centrosymmetric matrix is singular has been obtained, and two methods of computing inverse of the matrices(even order) have been given.
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Introduction
Centrosymmetric and anti-centrosymmetric matrices are two kinds of important special matrices, which are widely used in the fields of information theory, numerical analysis, linear system theory and so on. At present, some significant achievements have been acquired from the research on the structure and properties, eigenvalues and inverse of centrosymmetric and anti-centrosymmetric matrices. For example, in reference[3], Tan Ruimei used the definition of anticentrosymmetric matrix to prove some new conclusions on adjoint matrix, eigenvalue and eigenvector of it; in reference[6], Liu Lianfu discussed the method of computing inverse of anticentrosymmetric matrices in the light of the structure and representation of it. In this paper, based on the previous literatures, some systematic summary research has been made and some new proof methods have been shown for anti-centrosymmetric matrices.
Conclusion
Theorem 1
If A ∈ ACSRn×n, ξ is an eigenvector of A and λ0 is the corresponding eigenvalue, then −λ0 is also one of the eigenvalues of A. Besides, Jnξ is the eigenvector corresponding to −λ0.
Proof
From known results,
(λ0I − A) = 0
Because of Lemma 1,
Jn(λ0I − A)Jn = 0
and that is,
Jn(λ0I + A)Jn = 0, hence (λ0I + A) = 0
Then,
(−λ0I − A) = 0
and −λ0 is an eigenvalue of A.
Since
Aξ = λ0ξ
then
A(−ξ) = −λ0ξ
and according to Lemma 1,
JnAJnξ = −λ0ξ
Premultiplying both sides of the equation yields
JnJnAJnJnξ = −λ0ξ
From Definition 3,
A(Jnξ) = −λ0(Jnξ)
which shows that Jnξ is the eigenvector corresponding to −λ0.
Theorem 2
If A, B ∈ ACSRn×n, then kA ∈ ACSRn×n for any k ∈ ℂ, and A + B ∈ ACSRn×n. Thus, all n-order anti-centrosymmetric matrices over the complex field constitute a linear subspace of ℂn×n.
Theorem 3
The trace of an n-order anti-centrosymmetric matrix is 0, and therefore, the sum of the matrix's eigenvalues is 0.
Proof
In terms of Definition 2,
aij = −an−i+1, n−j+1, i, j = 1, 2, …, n
For odd-order anti-centrosymmetric matrices, the central element must be zero. Then the sum of the diagonal elements is
a11 + a22 + … + ann = 0
Because of all of the above, the trace of an n-order anti-centrosymmetric matrix is 0, and the sum of the eigenvalues is also 0.