Advances in Linear Algebra & Matrix Theory
Vol.3 No.4(2013), Article ID:40922,5 pages DOI:10.4236/alamt.2013.34010
Solution to a System of Matrix Equations
School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, China
Email: dongchangzh@sina.com, yuping.zh@163.com
Copyright © 2013 Changzhou Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received September 29, 2013; revised October 29, 2013; accepted November 5, 2013
Keywords: Matrix; Matrix Equation; Moore-Penrose Inverse; Approximation Problem; Least Squares Solution
ABSTRACT
A square complex matrix is called
if it can be written in the form
with
being fixed unitary and
being arbitrary matrix in
. We give necessary and sufficient conditions for the existence of the
solution to the system of complex matrix equation
and present an expression of the
solution to the system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least square
solution with least norm to this system mentioned above is considered. The representation of such solution is also derived.
1. Introduction
Throughout we denote the complex matrix space by
the real
matrix space by
The symbols
and
stand for the identity matrix with the appropriate size, the conjugate transpose, the range, the null space, and the Frobenius norm of
respectively. The Moore-Penrose inverse of
denoted by
is defined to be the unique matrix
of the following matrix equations
Recall that an complex matrix
is called
(or range Hermitian) if
matrices were introduced by Schwerdtfeger in [1], ever since many authors have studied
matrices with entries from complex number field to semigroups with involution and given various equivalent conditions and many characterizations for matrix to be
(see, [2-5]).
Investigating the matrix equation
(1)
with the unknown matrix being symmetric, reflexive, Hermitian-generalized Hamiltonian and re-positive definite is a very active research topic (see, [6-9]). As a generalization of (1), the classical system of matrix equations
(2)
has attracted many people’s attention and many results have been obtained about system (2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions, and so on (see, [9-12]). It is well-known that matrices are a wide class of objects that include many matrices as their special cases, such as Hermitian and skewHermitian matrices (i.e.,
), normal matrices (i.e.,
), as well as all nonsingular matrices. Therefore investigating the
solution of the matrix Equation (2) is very meaningful.
Pearl showed in ([2]) that a matrix is
if and only if it can be written in the form
with
unitary and
nonsingular. A square complex matrix
is called
if it can be written in the form
where
is fixed unitary and
is arbitrary matrix in
. To our knowledge, so far there has been little investigation of this
solution to (2).
Motivated by the work mentioned above, we investigate solution to (2). We also consider the optimal approximation problem
(3)
where is a given matrix in
and
the set of all
solutions to (2). In many case Equation (2) has not an
solution. Hence we need to further study its least squares solution, which can be described as follows: Let
denote the set of all
matrices with fixed unitary matrix
in
Find such that
(4)
In Section 2, we present necessary and sufficient conditions for the existence of the solution to (2), and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (3). In Section 4, we provide the least squares
solution to (4).
2. Solution to (2)
In this section, we establish the solvability conditions and the general expression for the solution to (2).
Throughout we denotes the set of all
matrices with fixed unitary matrix
in
i.e.,
where is fixed unitary and
is arbitrary matrix in
.
Lemma 2.1. ([3]) Let Then the system of matrix equations
is consistent if and only if
In that case, the general solution of this system is
where is arbitrary.
Now we consider the solution to (1). By the definition of
matrix, the solution has the following factorization:
Let
where
then (2) has
solution if and only if the system of matrix equations
is consistent. By Lemma 2.1, we have the following theorem.
Theorem 2.2. Let and
where
Then the matrix Equation (2) has a solution in
if and only if
(5)
In that case, the general solution of (1) is
(6)
where is arbitrary.
3. The Solution of Optimal Approximation Problem (3)
When the set of all
solution to (2) is nonempty, it is easy to verify
is a closed set. Therefore the optimal approximation problem (3) has a unique solution by [13]. We first verify the following lemma.
Lemma 3.1. Let Then the procrustes problem
has a solution which can be expressed as
where are arbitrary matrices.
Proof. It follows from the properties of Moore-Penrose generalized inverse and the inner product that
Hence,
if and only if
It is clear that with
are arbitrary is the solution of the above procrustes problem.
Theorem 3.2. Let and
(7)
where Assume
is nonempty, then the optimal approximation problem (3) has a unique solution
and
(8)
Proof. Since is nonempty,
has the form of (6). It follows from (7) and the unitary invariance of Frobenius norm that
Therefore, there exists such that the matrix nearness problem (3) holds if and only if exist
such that
According to Lemma 3.1, we have
where are arbitrary. Substituting
into (6), we obtain that the solution of the matrix nearness problem (3) can be expressed as (8).
4. The Least Squares Solution to (4)
In this section, we give the explicit expression of the least squares solution to (4).
Lemma 4.1. ([12]) Given
Then there exists a unique matrix
such that
And can be expressed as
where
Theorem 4.2. Let and
where,
Assume that the singular value decomposition of
are as follows
(9)
where
and
are unitary matrices,
,
Then
can be expressed as
(10)
where and
is an arbitrary matrix.
Proof. It yields from (9) that
Assume that
(11)
Then we have
Hence
is solvable if and only if there exist such that
(12)
(13)
It follows from (12) and (13) that
(14)
(15)
where Substituting (14) and (15)
into (11), we can get the form of elements in is (10).
Theorem 4.3. Assume the notations and conditions are the same as Theorem 4.2. Then
if and only if
(16)
where
Proof. In Theorem 4.2, it implies from (10) that
is equivalent to
has the expression (10)
with Hence (16) holds.
5. Acknowledgements
This research was supported by the Natural Science Foundation of Hebei province (A2012403013), the Natural Science Foundation of Hebei province (A2012205028) and the Education Department Foundation of Hebei province (Z2013110).
REFERENCES
- H. Schwerdtfeger, “Introduction to Linear Algebra and the Theory of Matrices,” P. Noordhoff, Groningen, 1950.
- M. H. Pearl, “On normal and
matrices,” Michigan Mathematical Journal, Vol. 6, No. 1, 1959, pp. 1-5. http://dx.doi.org/10.1307/mmj/1028998132
- C. R. Rao and S. K. Mitra, “Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971.
- O. M. Baksalary and G. Trenkler, “Characterizations of EP, normal, and Hermitian matrices,” Linear Multilinear Algebra, Vol. 56, 2008, pp. 299-304. http://dx.doi.org/10.1080/03081080600872616
- Y. Tian and H. X. Wang, “Characterizations of
Matrices and Weighted-EP Matrices,” Linear Algebra Applications, Vol. 434, No. 5, 2011, pp. 1295-1318. http://dx.doi.org/10.1016/j.laa.2010.11.014
- K.-W. E. Chu, “Singular Symmetric Solutions of Linear Matrix Equations by Matrix Decompositions,” Linear Algebra Applications, Vol. 119, 1989, pp. 35-50. http://dx.doi.org/10.1016/0024-3795(89)90067-0
- R. D. Hill, R. G. Bates and S. R. Waters, “On Centrohermitian Matrices,” SIAM Journal on Matrix Analysis and Applications, Vol. 11, No. 1, 1990, pp. 128-133. http://dx.doi.org/10.1137/0611009
- Z. Z. Zhang, X. Y. Hu and L. Zhang, “On the HermitianGeneralized Hamiltonian Solutions of Linear Mattrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 27, No. 1, 2005, pp. 294-303. http://dx.doi.org/10.1137/S0895479801396725
- A. Dajić and J. J. Koliha, “Equations
and
in Rings and Rings with Involution with Applications to Hilbert Space Operators,” Linear Algebra Applications, Vol. 429, No. 7, 2008, pp. 1779-1809. http://dx.doi.org/10.1016/j.laa.2008.05.012
- C. G. Khatri and S. K. Mitra, “Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 4, 1976, pp. 579-585. http://dx.doi.org/10.1137/0131050
- F. J. H. Don, “On the Symmetric Solutions of a Linear Matrix Equation,” Linear Algebra Applications, Vol. 93, 1987, pp. 1-7. http://dx.doi.org/10.1016/S0024-3795(87)90308-9
- H. X. Chang, Q. W. Wang and G. J. Song, “(R,S)-Conjugate Solution to a Pair of Linear Matrix Equations,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 73-82. http://dx.doi.org/10.1016/j.amc.2010.04.053
- E. W. Cheney, “Introduction to Approximation Theory,” McGraw-Hill Book Co., 1966.