The solvability conditions for left and right
inverse eigenvalue problems of
R-symmetric matrices
Yu-xia Du
∗ †
, Chuan-hua You, Kai-juan Shen
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, P. R. of China
Abstract
Let real matrix R be a symmetric and nontrivial involution, i.e. R = R
T
= R
−1
6= ±I
n
.
Real matrix A is called R -symmetric matrix if RAR = A. In this paper, the left and right
inverse eigenvalue problem for R-symmetric matrices is studied. We obtain the solvability
conditions and the solution expression of that. Furthermore, the optimal approximation
problem for given matrix A
∗
is also discussed. Finally, some numerical examples are given.
Our conclusions extend that of Fan-liang Li’s [Left and right inverse eigenpairs problem of
skew-centrosymmetric matrices, Appl. Math. Comput.177(2006)105-110].
Keywords: R-symmetric matrix; Left and right eigenpairs; Optimal approximation
1 Introduction
We firstly introduce some notations. Let R
n×m
be the set of all n × m real matrices and
OR
n×n
be the set of all n ×n orthogonal matrices. Denote by I
n
the identity matrix of order
n. And A
T
and A
†
represent the transpose and Moore-Penrose generalized inverse of matrix
A, respectively. The notation V
1
⊕V
2
stands for the orthogonal direct sum of linear subspace
V
1
and V
2
. For A = (a
ij
), B = (b
ij
) ∈ R
n×m
, (A, B) = tr(B
T
A) denotes the inner product of
matrices A and B, which generates Frobenius norm, i.e., kAk = (A, A)
1
2
= (tr(A
T
A))
1
2
.
Let real matrix R ∈ R
n×n
be a symmetric and nontrivial involution, i.e. R = R
T
=
R
−1
6= ±I
n
. A matrix A ∈ R
n×n
is said to be R-symmetric (R-antisymmetric) if RAR = A
(RAR = −A). The set of all R-symmetric (R-antisymmetric) matrices is denoted by RSR
n×n
(RASR
n×n
). In the following part, we always think that the matrix R is fixed.
In this paper, we discuss the left and right inverse eigenvalue problem of R-symmetric
matrices. We all know that the inverse eigenvalue problems has important applications
in mathematical modeling and parameter identification
[1]
. This problem has been studied
widely, see references [2-7] and therein. The so-called left and right inverse eigenvalue prob-
lem is: for given left and right eigenpairs (eigenvalue and its corresponding eigenvector),
∗
Corresponding author.
†
E-mail: xiaodoudou21@126.com (Y.X. Du)
1
http://www.paper.edu.cn