Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C ()
1. Introduction
System theory, stochastic control, and differential methods for solving elliptic partial differential equations frequently involve the class of integer-order nonlinear equations known as algebraic Riccati matrix equations [1] [2] [3] [4].
In recent years, many scholars have extended Riccati matrix equations to matrix equations of non-integer order or symmetric structure type, and have achieved rich results. For example, [5]-[13] studied various iterative methods for Hermite positive definite solutions of nonlinear matrix equations of the form
or
for different values of the unknown matrix indices
; in [14] [15] [16] [17], the existence and perturbation analysis of Hermite positive definite solutions of matrix equation
are given. [18] [19] discussed the extremal solutions of matrix equation
and the upper and lower bounds of solutions. However, the equations mentioned above are instead discussed in the real number field or the complex number field. Research on the solutions of quaternion nonlinear matrix equations is uncommon [20], particularly for some structural solutions of non-integer order quaternion matrix equations, there is currently no relevant research report.
This paper focuses on the quaternion matrix equation
(1)
study its Hermite positive definite solution, where
is a positive number,
and
(Hermite positive definite, hereafter referred to as positive definite) is the known quaternion matrix,
is an unknown matrix. To discuss convenience, let
and
note the conjugate, conjugate transpose and respectively of the quaternion matrix
. The matrix
is classified as Hermite or self-conjugate if
. All n-order Hermite matrices are classified as
. The
are used to represent the maximum and minimum eigenvalues of the Hermite matrix
,
respectively denotes the Frobenius norm of the quaternion matrix
. For n-order positive definite matrices
, the mean by
that
is positive definite. For quaternion matrix
We call
is the complexization operator of
[21]. The complexification operator concerning a quaternionic matrix
has the following operational properties [22]
2. Existence of Positive Definite Solutions
This section first discusses some necessary and sufficient conditions for the existence of Hermite positive definite solutions to the quaternion matrix Equation (1). For the coefficient matrix in (1), denote
(2)
then, there are the following conclusions.
Theorem 1. Let
,
is a nonsingular matrix,
is a positive definite matrix. The real numbers
are given by (2), if the following system of algebraic equations
(3)
for
has positive real pairs of solutions
, then there must exist a Hermite positive definite solution to the matrix Equation (1).
Proof. Let
be a positive real pair solution to the system of algebraic Equation (3). The existence of Hermite positive definite solutions of matrix Equation (1) is discussed below in three cases.
Case 1:
. It can be obtained from (3)
(4)
Because of
, therefore, Equation (4) holds if and only if
(5)
According to (5) and the properties of self-conjugate quaternion matrices, it follows that
(6)
Then,
is a Hermite positive definite solution of (1).
Case 2:
. At this point, write that
, then
is a nonempty bounded closed convex set.
, we obtain
(7)
Thus, from (7) and in connection with (3), we have
(8)
Therefore, a matrix function can be constructed on
(9)
This
is continuous
, and by (8) we get
(10)
Thus
, according to the Brouwer fixed point theorem,
must have a fixed point on
, that is, the matrix Equation (1) has a Hermite positive definite solution.
Case 3:
. At this point, note that
, then
is a nonempty bounded closed convex set.
, we obtain
(11)
Thus from (11) and in connection with (3) we gain
(12)
Again by
nonsingular and (12)
(13)
Therefore, a matrix function
(14)
can be constructed on
.
This
is continuous
, and by (13) we get
Thus
, by the Brouwer fixed point theorem,
must have a fixed point on
, that is, the matrix Equation (1) has a Hermite positive definite solution.
The following discussion further explores the conditions under which the real number
in Theorem 1 satisfies, leading to the existence of a solution
for positive real number pairs in an algebraic equation system (3). In this regard, the following results are given.
Corollary 1. If the real number
in (2) satisfies one of the following conditions
(a)
(b)
then, the system of algebraic Equation (3) has positive real pair solutions.
Proof. (a) It is easy to know that the system of Equation (3) is equivalent to the following system of equations.
(15)
we write
(16)
It is obvious that
is continuous in
. When the condition (a) is satisfied, we can obtain
So we get
(17)
Therefore, a positive real number
exists such that
and
, that is, the system of algebraic Equation (3) has a positive real pair solution in
.
(b) When this condition holds, it follows that
Therefore, in the closed interval
, the function (16) is also used to obtain
(18)
So, the system of algebraic Equation (3) has a positive real pair solution
in
.
In Equation (1), when
(identity matrix),
(quaternion unitary matrix), we obtain the following corollary.
Corollary 2. Let
be a quaternion unitary matrix, then
, the matrix equation
always exists Hermite positive definite solutions.
Proof When
,
,
in (2), then, the system of Equation (3) becomes
(19)
It is easy to know that the system of Equation (19) has a positive real number pair solution
in
. Based on Theorem 1, the conclusion holds. Using the symmetry of the system of Equation (19), it is known that
is the unique solution of the equation
in
, it follows that
is a positive definite solution of the given equation.
Corollary 3. Let
, If
satisfies
. (20)
Then, the matrix equation
exists a Hermite positive definite solution.
Proof. When
,
in (2), hence the system of Equation (3) becomes
(21)
It is evident that the function
is continuous on the interval
, and when condition (20) holds, we gain
So
has a real root
in
, and thus
, It follows from Theorem 1 that, the given equation has a Hermite positive definite solution.
Corollary 4. Let
, then the necessary and sufficient condition for the matrix Equation (1) to have a positive definite solution
is that the matrix
has the following decomposition
(22)
where
is a column unitary orthogonal matrix.
Proof. (Necessity) If
is a positive definite solution of the equation
, such that
(23)
We define
, then
, from (23), we know that
is a column unitary orthogonal matrix.
(Sufficiency) If the matrix
has the decomposition (22), where
is a positive definite matrix and
Substituting
into the left side of Equation (1), we get
. (24)
It can be seen that
is a positive definite solution to the equation
.
The upper and lower bound estimates for the positive definite solution of Equation (1) are given below.
Theorem 2. Let Equation (1) have a positive definite solution
, and
,
the
(25)
Proof. Let
be a positive definite solution of Equation (1), such that
(26)
Furthermore, we have
, therefore
(27)
It can be seen from (26) and (27) that (25) is established.
Corollary 5. If the real numbers in (2) satisfy
, then Equation (1) must have a positive definite solution
, and
(28)
Proof. By
, based on Theorem 1 and Corollary 1(a), Equation (1) exists a positive definite solution
. Then by the proof method of Theorem 2, we obtained
(29)
and there
(30)
It can be seen from (29) and (30) that (28) is established.
Theorem 3. Let Equation (1) exist positive definite solutions
,
nonsingular and
, then
(31)
Proof. Let
be a positive definite solution of Equation (1). Then by
non-singularity, we gain
(32)
Moreover, from the condition
, so
, it can be obtained from (32)
thus have
(33)
From (32) and (33), it can be seen that (31) is established.
Corollary 6. Under the condition of Theorem 3, if the real number in (2) satisfy
, then Equation (1) must have a positive definite solution
, and
(34)
Proof. By
, According to Theorem 1 and Corollary 1, there exists a positive definite solution
to Equation (1). Then by the proof method of Theorem 3, we get
(35)
therefore, there is
then
(36)
According to (35) and (36), (34) holds.
3. Iterative Method of Positive Definite Solution
It follows from the results presented in Section 2 that, when the coefficient matrices
of Equation (1) satisfy the given conditions, we can construct the positive definite solution iteration scheme of Equation (1).
(I) Under the condition of Theorem 1, when
, the iterative scheme is established as follows:
(37)
(II) Under the condition of Theorem 1, when
, the iterative scheme is established as follows:
(38)
(III) Under the condition of Theorem 2 and Corollary 5, when
, the iterative scheme is established as follows:
(39)
(IV) Under the condition of Theorem 3 and Corollary 6, when
, the iterative scheme is established as follows:
(40)
where
denotes the complex operator of the quaternion matrix
. In the actual calculation, due to the non-commutative reason of quaternion multiplication, we use the right iterative format above (37), (38), (39), (40) to calculate in the Matlab software. Finally, the kth approximate solution
is reduced back to
, which is the approximate solution of Equation (1). According to the relation between a quaternion matrix and the Frobenius norm of its complex representation matrix, the residual term norm of the first approximate solution of Equation (1) is
Note: The selection of the initial matrix of the iteration.
(I) From Theorem 2 and Corollary 5, when the condition
or the real numbers in (2) satisfy
, the initial matrix can be selected as
, or
. At this time, it is not necessary to solve
in Equation (3).
(II) From Theorem 3 and Corollary 6, when the condition
is nonsingular, and the real numbers in
or (2) satisfy
, the initial matrix can be selected as
, or
. At this time, it is also not necessary to solve
in Equation (3).
4. Numerical Examples
This section mainly focuses on the characteristics of the coefficient matrix of Equation (1), combined with the iterative formula constructed in Section 3. It uses numerical examples to illustrate the effectiveness and feasibility of the results in this paper. Firstly, based on the results of Theorem 1, two examples are used to illustrate that the iterative formula can be effectively used to calculate the positive definite solution of the equation when the equation satisfies the given conditions.
Example 4.1. Let
, considering the quaternion matrix Equation (1), two n-order quaternion matrices are given as follows:
Establish an appropriate iterative scheme to find the positive definite solution of Equation (1)
Solution: From the complex decomposition formula
of the quaternion matrix, we can get
Direct calculation shows that
Then, using Theorem 1, Equation (1) has a positive definite solution, which is solved by iteration (37). when
, The results of the iterations are shown in Table 1.
Example 4.2. Let
, Considering the quaternion matrix Equation (1), two n-order quaternion matrices are given as follows:
Table 1. Iterative calculation results under different matrix orders
Solution: From the complex decomposition formula
of the quaternion matrix, we can get
Direct calculation shows that
As inequality
holds, From Theorem 2, an Equation (1) has a positive definite solution, which is solved by iteration (39). when
, The results of the iterations are shown in Table 2.
From the comparison of the above two examples, it can be seen that when the inequality
holds, the corresponding iteration has a smaller CPU running time.
5. Conclusion
The criteria for the existence of Hermite positive definite solutions of a class of nonlinear matrix Equation (1) on the quaternion field and the iterative solution method are given. The maximum and minimum eigenvalues of matrices
and
are mainly used to create the system of algebraic Equation (3) and
Table 2. Iterative calculation results under different matrix orders
through discussion, the existence of its positive real number solutions, Obtaining some necessary and sufficient conditions for the existence of Hermite positive definite solutions of matrix equations, thus, the existence interval of the solution and the upper and lower bounds estimation formula of the solution are obtained. At the same time, according to the size relationship of the positive real solution
of the equation group (3), the iterative (37), (38), (39), (40) of convergence are constructed respectively. Two numerical examples verify the effectiveness and feasibility of the given iteration. It is concluded that when the eigenvalues satisfy the inequality
, the corresponding iterative formula has a higher convergence speed. The results can be effectively used to judge and calculate the Hermitian positive definite solution of the quaternion matrix Equation (1), which extends the problem of solving non-integer order matrix equations in complex fields.
Acknowledgements
I would like to thank professor Jingpin Huang for his patient and professional guidance during the dissertation, and to appreciate all my family and friends who are caring, encouraging, and supporting me. In addition, thanks to the national natural science foundation of China (no.12361078) for supporting this work.