Gradient-based iterative algorithm is suggested for solving a coupled complex conjugate and transpose matrix equations. Using the hierarchical identification principle and the real representation of a complex matrix, a convergence proof is offered. The necessary and sufficient conditions for the optimal convergence factor are determined. A numerical example is offered to validate the efficacy of the suggested algorithm.
How to solve a matrix equation is a focus question of the applied mathematics and engineering fields [
Although achieving some fruitful results, there are still some problems that need to be solved. For example, the gradient-based iterative algorithm for solving a class of complex matrix equations was established and the sufficient condition for the convergence of the algorithm was obtained [
Inspired by the above work, this paper discusses gradient-based iterative algorithm for a coupled complex conjugate and transpose matrix equation. Using the real representation of a complex matrix, the convergence proof is given and the optimal convergence factor is determined. An numerical test is offered to validate the efficacy of the suggested algorithm.
We organized this paper as follows. Some preliminary lemmas and symbols are listed in Section 2. Section 3 proves convergence of the gradient-based iterative algorithm for solving a coupled complex conjugate and transpose matrix equation. Section 4 brings a numerical test to validate the efficacy of the proposed algorithm. Some concluding remarks are given in Section 5.
Let A be an m × n complex matrix, symbols A ¯ , A H , A T , λ m a x [ A ] , λ m i n [ A ] , ‖ A ‖ 2 , ‖ A ‖ represents the conjugate, the conjugate transpose, the transpose, the maximal eigenvalue, the minimal nonzero eigenvalue, the spectral norm, the Frobenius norm, respectively. I n denotes an identity matrix of size n × n . Let A be m × n real matrix and rank [ A ] = r , symbol σ i ( A ) , i = 1,2, ⋯ , r represents the nonzero singular values that satisfy σ 1 ( A ) ≥ σ 2 ( A ) ≥ ⋯ ≥ σ r ( A ) . col [ X ] denotes an mn-dimensional vector defined by the formula col [ X ] = [ x 1 T , x 2 T , ⋯ , x n T ] T . Here,
X = [ x 1 , x 2 , ⋯ , x n ] ∈ ℝ m × n . The Kronecker product A ⊗ B of matrices A and B is defined by formula A ⊗ B = [ a i j B ] ∈ ℝ ( m p ) × ( n q ) . Here A = ( a i j ) m × n , B = ( b i j ) p × q .
Next lemma is a conclusion of the singular values decomposition of a complex matrix, the proof can be found in a standard matrix theory book, we omit the proof here.
Lemma 1. A is an m × n complex matrix and rank [ A ] = r . There exist two unitary matrices U ∈ ℂ n × n and V ∈ ℂ m × m such that
V H A U = [ Σ 0 0 0 ] , (1)
where Σ = diag { σ 1 , σ 2 , ⋯ , σ r } with order σ 1 ≥ σ 2 ≥ ⋯ ≥ σ r > 0 .
Refer to [
For a complex matrix A ∈ ℂ m × n , it can be exclusive signified as A = A 1 + i A 2 , where A 1 , A 2 ∈ ℝ m × n are two real matrices. Now we use symbol A σ to represent it as
A σ : = [ A 1 A 2 A 2 − A 1 ] ∈ ℝ 2 m × 2 n . (2)
with this symbol, in the following we use the following notation A ¯ σ : = ( A ¯ ) σ , A σ H : = ( A H ) σ and A σ − 1 : = ( A σ ) − 1 . Because
( A T ) σ = ( A 1 T + i A 2 T ) σ = [ A 1 T A 2 T A 2 T − A 1 T ] = [ A 1 A 2 A 2 − A 1 ] T = ( A σ ) T ,
we use A σ T to represent ( A T ) σ and have symbols:
{ A σ T : = ( A T ) σ , A ¯ σ : = ( A ¯ ) σ , A σ H : = ( A H ) σ , A σ − 1 : = ( A σ ) − 1 . (3)
Using symbols [
P m : = [ I m 0 0 − I m ] , Q m : = [ 0 I m − I m 0 ] ,
We have the following lemma [
Lemma 2. The real representation possesses the following properties:
1) If A , B ∈ ℂ m × n , a ∈ ℝ , then
{ ( A + B ) σ = A σ + B σ , ( a A ) σ = a A σ , ( A σ T ) T = A σ .
2) If A ∈ ℂ m × n , then
{ A σ = Q m A σ Q n , A ¯ σ = P m A σ P n , A σ H = P m A σ T P n .
3) If A ∈ ℂ m × n , B ∈ ℂ n × r , C ∈ ℂ r × p , then
{ ( A B ) σ = A σ P n B σ = A σ B ¯ σ P r , ( A B C ) σ = A σ B ¯ σ C σ .
The proof one can refer to [
Next lemma describes the properties of the spectral norm and the Frobenius norm.
Lemma 3. Let A be a complex matrix, we have
1) ‖ A σ ‖ 2 = 2 ‖ A ‖ 2 ,
2) ‖ A σ ‖ 2 = ‖ A ‖ 2 .
For the complex matrix equation A X B = F , referring to [
Lemma 4. For the complex matrix equation A X B = F , if A is a column-full rank matrix and B is a row-full rank matrix, then the iterative solution X ( k ) given by the gradient-based iterative algorithm
X ( k ) = X ( k − 1 ) + μ A H [ F − A X ( k − 1 ) B ] B H
converges to the exact solution X * (i.e., l i m k → ∞ X ( k ) = X * ) for any initial values X ( 0 ) if and only if
0 < μ < 2 λ m a x [ A H A ] λ m a x [ B B H ] .
Moreover, the best convergence factor μ 0 is
μ 0 = 2 λ m a x [ A H A ] λ m a x [ B B H ] + λ m i n [ A H A ] λ m i n [ B B H ] .
In this section, we suggest the gradient-based iterative algorithm for solving the complex coupled conjugate and transpose matrix equations. For convenience, we introduce the following operator
L i j ( X j ) : = A i j X j B i j + C i j X ¯ j D i j + G i j X j T H i j + M i j X j H N i j , i , j = 1,2. (4)
Using these operators, we consider the following complex coupled matrix equations
{ L 11 ( X 1 ) + L 12 ( X 2 ) = F 1 , L 21 ( X 1 ) + L 22 ( X 2 ) = F 2 , (5)
where A i j , B i j , C i j , D i j , G i j , H i j , M i j , N i j , F i ∈ ℂ n × n , i , j = 1,2 , are known matrices and X 1 , X 2 ∈ ℂ n × n are the matrices to be determined.
According to the definition of the real representation of a complex matrix and Lemma 2, the real representation of matrix ( L i j ( X j ) ) σ denoted as L i j σ ( X j ) can be rewritten as
L i j σ ( X j ) : = ( L i j ( X j ) ) σ = ( A i j X j B i j + C i j X ¯ j D i j + G i j X j T H i j + M i j X j H N i j ) σ = ( A i j X j B i j ) σ + ( C i j X ¯ j D i j ) σ + ( G i j X j T H i j ) σ + ( M i j X j H N i j ) σ = A i j σ X ¯ j σ B i j σ + C i j σ X j σ D i j σ + G i j σ X j σ H H i j σ + M i j σ X j σ T N i j σ = A i j σ P n X j σ P n B i j σ + C i j σ X j σ D i j σ + G i j σ P n X j σ T P n H i j σ + M i j σ X j σ T N i j σ . (6)
Taking the vec-operator of the above equation gives
col [ L i j σ ( X j ) ] = col [ A i j σ P n X j σ P n B i j σ ] + col [ C i j σ X j σ D i j σ ] + col [ G i j σ P n X j σ T P n H i j σ ] + col [ M i j σ X j σ T N i j σ ] = ( ( P n B i j σ ) T ⊗ A i j σ P n ) col [ X j σ ] + ( D i j σ T ⊗ C i j σ ) col [ X j σ ] + ( ( P n H i j σ ) T ⊗ G i j σ P n ) col [ X j σ T ] + ( N i j σ T ⊗ M i j σ ) col [ X j σ T ]
= ( ( P n B i j σ ) T ⊗ A i j σ P n ) col [ X j σ ] + ( D i j σ T ⊗ C i j σ ) col [ X j σ ] + ( ( P n H i j σ ) T ⊗ G i j σ P n ) P ( 2 n ,2 n ) col [ X j σ ] + ( N i j σ T ⊗ M i j σ ) P ( 2 n ,2 n ) col [ X j σ ] = Φ i j σ col [ X j σ ] . (7)
Here, symbol Φ i j σ is defined as
Φ i j σ : = ( ( P n B i j σ ) T ⊗ A i j σ P n ) + ( ( P n H i j σ ) T ⊗ G i j σ P n ) P ( 2 n ,2 n ) + ( D i j σ T ⊗ C i j σ ) + ( N i j σ T ⊗ M i j σ ) P ( 2 n ,2 n ) . (8)
Taking the real representation of Equation (5) gives
{ L 11 σ ( X 1 ) + L 12 σ ( X 2 ) = F 1 σ , L 21 σ ( X 1 ) + L 22 σ ( X 2 ) = F 2 σ , (9)
Taking the vec-operator of the above system, we have
{ col [ L 11 σ ( X 1 ) ] + col [ L 12 σ ( X 2 ) ] = col [ F 1 σ ] , col [ L 21 σ ( X 1 ) ] + col [ L 22 σ ( X 2 ) ] = col [ F 2 σ ] . (10)
Using notation Φ i j σ gives
{ Φ 11 σ col [ X 1 σ ] + Φ 12 σ col [ X 2 σ ] = col [ X 1 σ ] , Φ 21 σ col ( X 1 σ ) + Φ 22 σ col [ X 2 σ ] = col [ X 2 σ ] . (11)
Setting
Φ σ : = [ Φ 11 σ Φ 12 σ Φ 21 σ Φ 22 σ ] ,
Equation (11) can be rewritten in a compact one as
Φ σ col [ X 1 σ , X 2 σ ] = col [ F 1 σ , F 2 σ ] . (12)
For Equation (12), we have the following result.
Lemma 5. If matrix Φ σ is invertible, then Equation (12) has a unique solution and it can be given by formula
col [ F 1 σ , F 2 σ ] = Φ σ − 1 col [ X 1 σ , X 2 σ ] .
According to Lemma 4, the gradient-based iterative algorithm
col [ X 1 σ ( k + 1 ) , X 2 σ ( k + 1 ) ] = ( I 8 n 2 − μ Φ σ T Φ σ ) col [ X 1 σ ( k ) , X 2 σ ( k ) ] + μ Φ σ T col [ X 1 σ , X 2 σ ] (13)
converges to the exact solution pair col [ X 1 ∗ ] and col [ X 2 ∗ ] for any initial values X 1 ( 0 ) and X 2 ( 0 ) if and only if
0 < μ < 2 λ m a x [ Φ σ T Φ σ ] .
Moreover, the best convergence factor μ 0 is
μ 0 = 2 λ m a x [ Φ σ T Φ σ ] + λ m i n [ Φ σ T Φ σ ] .
Clearly, if X 1 , X 2 ∈ ℂ n × n then we have L i j σ ∈ ℝ 2 n × 2 n , Φ i j σ ∈ ℝ 4 n 2 × 4 n 2 and Φ ∈ ℝ 8 n 2 × 8 n 2 . The increasing dimension leads to a computational difficulty due to an excessive computer memory. To overcome this problem, next we construct a gradient-based iterative algorithm by using the hierarchical identification principle.
For convenience, we introduce notation Ψ 1 and Ψ 2 as follows:
Ψ 1 : = F 1 − L 11 ( X 1 ( k ) ) − L 12 ( X 2 ( k ) ) , (14)
Ψ 2 : = F 2 − L 21 ( X 1 ( k ) ) − L 22 ( X 2 ( k ) ) . (15)
According to the hierarchical identification principle, we introduce the intermediate matrices:
F 11 : = Ψ 1 + A 11 X 1 B 11 , F 12 : = Ψ 1 + C 11 X ¯ 1 D 11 , (16)
F 13 : = Ψ 1 + G 11 X 1 T H 11 , F 14 : = Ψ 1 + M 11 X 1 H N 11 , (17)
F 15 : = Ψ 1 + A 12 X 2 B 12 , F 16 : = Ψ 1 + C 12 X ¯ 2 D 12 , (18)
F 17 : = Ψ 1 + G 12 X 2 T H 12 , F 18 : = Ψ 1 + M 12 X 2 H N 12 , (19)
F 21 : = Ψ 2 + A 21 X 1 B 21 , F 22 : = Ψ 2 + C 21 X ¯ 1 D 21 , (20)
F 23 : = Ψ 2 + G 21 X 1 T H 21 , F 24 : = Ψ 2 + M 21 X 1 H N 21 , (21)
F 25 : = Ψ 2 + A 22 X 2 B 22 , F 26 : = Ψ 2 + C 22 X ¯ 2 D 22 , (22)
F 27 : = Ψ 2 + G 22 X 2 T H 22 , F 28 : = Ψ 2 + M 22 X 2 H N 22 . (23)
Equation (5) can be decomposed into the following matrix equations
A 11 X 1 B 11 = F 11 , C 11 X ¯ 1 D 11 = F 12 , (24)
G 11 X 1 T H 11 = F 13 , M 11 X 1 H N 11 = F 14 , (25)
A 12 X 2 B 12 = F 15 , C 12 X ¯ 2 D 12 = F 16 , (26)
G 12 X 2 T H 12 = F 17 , M 12 X 2 H N 12 = F 18 , (27)
A 21 X 1 B 21 = F 21 , C 21 X ¯ 1 D 21 = F 22 , (28)
G 21 X 1 T H 21 = F 23 , M 21 X 1 H N 21 = F 24 , (29)
A 22 X 2 B 22 = F 25 , C 22 X ¯ 2 D 22 = F 26 , (30)
G 22 X 2 T H 22 = F 27 , M 22 X 2 H N 22 = F 28 . (31)
According to Lemma 4, with proper manipulations, one can construct the following iterative algorithms for matrix equations in Equations (24)-(30) respectively,
X 11 ( k + 1 ) = X 11 ( k ) + μ A 11 H ( F 11 − A 11 X 11 ( k ) B 11 ) B 11 H , (32)
X 12 ( k + 1 ) = X 12 ( k ) + μ C 11 T ( F ¯ 12 − C ¯ 11 X 12 ( k ) D ¯ 11 ) D 11 T , (33)
X 13 ( k + 1 ) = X 13 ( k ) + μ H ¯ 11 ( F 13 T − H 11 T X 13 ( k ) G 11 T ) G ¯ 11 , (34)
X 14 ( k + 1 ) = X 14 ( k ) + μ N 11 ( F 14 H − N 11 H X 14 ( k ) M 11 H ) M 11 , (35)
X 21 ( k + 1 ) = X 21 ( k ) + μ A 12 H ( F 15 − A 12 X 21 ( k ) B 12 ) B 12 H , (36)
X 22 ( k + 1 ) = X 22 ( k ) + μ C 12 T ( F ¯ 16 − C ¯ 12 X 22 ( k ) D ¯ 12 ) D 12 T , (37)
X 23 ( k + 1 ) = X 23 ( k ) + μ H ¯ 12 ( F 17 T − H 12 T X 23 ( k ) G 12 T ) G ¯ 12 , (38)
X 24 ( k + 1 ) = X 24 ( k ) + μ N 12 ( F 18 H − N 12 H X 24 ( k ) M 12 H ) M 12 , (39)
X 15 ( k + 1 ) = X 15 ( k ) + μ A 21 H ( F 21 − A 21 X 15 ( k ) B 21 ) B 21 H , (40)
X 16 ( k + 1 ) = X 16 ( k ) + μ C 21 T ( F ¯ 22 − C ¯ 21 X 16 ( k ) D ¯ 21 ) D 21 T , (41)
X 17 ( k + 1 ) = X 17 ( k ) + μ H ¯ 21 ( F 23 T − H 21 T X 17 ( k ) G 21 T ) G ¯ 21 , (42)
X 18 ( k + 1 ) = X 18 ( k ) + μ N 21 ( F 24 H − N 21 H X 18 ( k ) M 21 H ) M 21 , (43)
X 25 ( k + 1 ) = X 25 ( k ) + μ A 22 H ( F 25 − A 22 X 25 ( k ) B 22 ) B 22 H , (44)
X 26 ( k + 1 ) = X 26 ( k ) + μ C 22 T ( F ¯ 26 − C ¯ 22 X 26 ( k ) D ¯ 22 ) D 22 T , (45)
X 27 ( k + 1 ) = X 27 ( k ) + μ H ¯ 22 ( F 27 T − H 22 T X 27 ( k ) G 22 T ) G ¯ 21 , (46)
X 28 ( k + 1 ) = X 28 ( k ) + μ N 22 ( F 28 H − N 22 H X 28 ( k ) M 22 H ) M 22 . (47)
Substituting Equations (16)-(23) into Equations (32)-(47) respectively gives
X 11 ( k + 1 ) = X 11 ( k ) + μ A 11 H ( Ψ 1 + A 11 X 1 B 11 − A 11 X 11 ( k ) B 11 ) B 11 H ,
X 12 ( k + 1 ) = X 12 ( k ) + μ C 11 T ( Ψ ¯ 1 + C ¯ 11 X 1 D ¯ 11 − C ¯ 11 X 12 ( k ) D ¯ 11 ) D 11 T ,
X 13 ( k + 1 ) = X 13 ( k ) + μ H ¯ 11 ( Ψ 1 T + H 11 T X 1 G 11 T − H 11 T X 13 ( k ) G 11 T ) G ¯ 11 ,
X 14 ( k + 1 ) = X 14 ( k ) + μ N 11 ( Ψ 1 H + N 11 H X 1 M 11 H − N 11 H X 14 ( k ) M 11 H ) M 11 ,
X 21 ( k + 1 ) = X 21 ( k ) + μ A 12 H ( Ψ 1 + A 12 X 2 B 12 − A 12 X 21 ( k ) B 12 ) B 12 H ,
X 22 ( k + 1 ) = X 22 ( k ) + μ C 12 T ( Ψ ¯ 1 + C ¯ 12 X 2 D ¯ 12 − C ¯ 12 X 22 ( k ) D ¯ 12 ) D 12 T ,
X 23 ( k + 1 ) = X 23 ( k ) + μ H ¯ 12 ( Ψ 1 T + H 12 T X 2 G 12 T − H 12 T X 23 ( k ) G 12 T ) G ¯ 12 ,
X 24 ( k + 1 ) = X 24 ( k ) + μ N 12 ( Ψ 1 H + N 12 H X 2 M 12 H − N 12 H X 24 ( k ) M 12 H ) M 12 ,
X 15 ( k + 1 ) = X 15 ( k ) + μ A 21 H ( Ψ 2 + A 21 X 1 B 21 − A 21 X 15 ( k ) B 21 ) B 21 H ,
X 16 ( k + 1 ) = X 16 ( k ) + μ C 21 T ( Ψ ¯ 2 + C ¯ 21 X 1 D ¯ 21 − C ¯ 21 X 16 ( k ) D ¯ 21 ) D 21 T ,
X 17 ( k + 1 ) = X 17 ( k ) + μ H ¯ 21 ( Ψ 2 T + H 21 T X 1 G 21 T − H 21 T X 17 ( k ) G 21 T ) G ¯ 21 ,
X 18 ( k + 1 ) = X 18 ( k ) + μ N 21 ( Ψ 2 H + N 21 H X 1 M 21 H − N 21 H X 18 ( k ) M 21 H ) M 21 ,
X 25 ( k + 1 ) = X 25 ( k ) + μ A 22 H ( Ψ 2 + A 22 X 2 B 22 − A 22 X 25 ( k ) B 22 ) B 22 H ,
X 26 ( k + 1 ) = X 26 ( k ) + μ C 22 T ( Ψ ¯ 2 + C ¯ 22 X 2 D ¯ 22 − C ¯ 22 X 26 ( k ) D ¯ 22 ) D 22 T ,
X 27 ( k + 1 ) = X 27 ( k ) + μ H ¯ 22 ( Ψ 2 T + H 22 T X 2 G 22 T − H 22 T X 27 ( k ) G 22 T ) G ¯ 21 ,
X 28 ( k + 1 ) = X 28 ( k ) + μ N 22 ( Ψ 2 H + N 22 H X 2 M 22 H − N 22 H X 28 ( k ) M 22 H ) M 22 .
The unknown matrices X 1 and X 2 appear in the right-hand sides of the above equations, so it is impossible to realize the above algorithms. According to the hierarchical identification principle, the unknown matrices X 1 and X 2 can be replaced by their estimates X 1 ( k ) and X 2 ( k ) . Since only two iterative values X 1 ( k ) and X 2 ( k ) are needed, we take the average of X 11 ( k ) , X 12 ( k ) , X 13 ( k ) , X 14 ( k ) , X 15 ( k ) , X 16 ( k ) , X 17 ( k ) and X 18 ( k ) as the iterative value X 1 ( k ) and X 21 ( k ) , X 22 ( k ) , X 23 ( k ) , X 24 ( k ) , X 25 ( k ) , X 26 ( k ) , X 27 ( k ) and X 28 ( k ) as the iterative value X 2 ( k ) , respectively. We have the following algorithm:
Ψ 1 = F 1 − L 11 ( X 1 ( k ) ) − L 12 ( X 2 ( k ) ) , (48)
Ψ 2 = F 2 − L 21 ( X 1 ( k ) ) − L 22 ( X 2 ( k ) ) , (49)
X 11 ( k + 1 ) = X 1 ( k ) + μ A 11 H Ψ 1 B 11 H , (50)
X 12 ( k + 1 ) = X 1 ( k ) + μ C 11 T Ψ ¯ 1 D 11 T , (51)
X 13 ( k + 1 ) = X 1 ( k ) + μ H ¯ 11 Ψ 1 T G ¯ 11 , (52)
X 14 ( k + 1 ) = X 1 ( k ) + μ N 11 Ψ 1 H M 11 , (53)
X 21 ( k + 1 ) = X 2 ( k ) + μ A 12 H Ψ 1 B 12 H , (54)
X 22 ( k + 1 ) = X 2 ( k ) + μ C 12 T Ψ ¯ 1 D 12 T , (55)
X 23 ( k + 1 ) = X 2 ( k ) + μ H ¯ 12 Ψ 1 T G ¯ 12 , (56)
X 24 ( k + 1 ) = X 2 ( k ) + μ N 12 Ψ 1 H M 12 , (57)
X 15 ( k + 1 ) = X 1 ( k ) + μ A 21 H Ψ 2 B 21 H , (58)
X 16 ( k + 1 ) = X 1 ( k ) + μ C 21 T Ψ ¯ 2 D 21 T , (59)
X 17 ( k + 1 ) = X 1 ( k ) + μ H ¯ 21 Ψ 2 T G ¯ 21 , (60)
X 18 ( k + 1 ) = X 1 ( k ) + μ N 21 Ψ 2 H M 21 , (61)
X 25 ( k + 1 ) = X 2 ( k ) + μ A 22 H Ψ 2 B 22 H , (62)
X 26 ( k + 1 ) = X 2 ( k ) + μ C 22 T Ψ ¯ 2 D 22 T , (63)
X 27 ( k + 1 ) = X 2 ( k ) + μ H ¯ 22 Ψ 2 T G ¯ 21 , (64)
X 28 ( k + 1 ) = X 2 ( k ) + μ N 22 Ψ 2 H M 22 . (65)
X 1 ( k ) = 1 8 ∑ p = 1 8 X 1 p ( k ) , (66)
X 2 ( k ) = 1 8 ∑ q = 1 8 X 2 q ( k ) . (67)
Theorem 6. If the complex matrix Equation (5) has a unique solution pair X 1 * and X 2 * , then the iterative solution pair X 1 ( k ) and X 2 ( k ) given by the algorithm in Equations (50)-(66), converges to the unique solution pair X 1 * and X 2 * if and only if
0 < μ < 16 λ max 2 [ Φ σ ] = : μ max . (68)
Under these conditions, the best convergence factor μ 0 is
μ 0 = 16 λ m a x 2 [ Φ σ ] + λ m i n 2 [ Φ σ ] . (69)
Proof. Similar to symbols in Equation (3), we use the following notation:
{ Ψ 1 σ : = ( Ψ 1 ) σ , Ψ 2 σ : = ( Ψ 2 ) σ , L i j σ T ( X ) : = ( L i j T ( X ) ) σ , L i j σ ( X ) ¯ : = ( L i j ( X ) ¯ ) σ , L i j σ H ( X ) : = ( L i j H ( X ) ) σ . (70)
For convenience, we set X j σ ( k ) : = ( X j ( k ) ) σ . Taking the real representation of the both sides of Equation (50) gives
X 11 σ ( k + 1 ) = X 1 σ ( k ) + μ ( A 11 H Ψ 1 B 11 H ) σ = X 1 σ ( k ) + μ A 11 σ H ( Ψ 1 ¯ ) σ B 11 σ H = X 1 σ ( k ) + μ A 11 σ H P n Ψ 1 σ P n B 11 σ H = X 1 σ ( k ) + μ P n A 11 σ T Ψ 1 σ B 11 σ T P n . (71)
Similarly, taking the real representation of Equations (50)-(64) give
X 12 σ ( k + 1 ) = X 1 σ ( k ) + μ C 11 σ T Ψ 1 σ D 11 σ T , (72)
X 13 σ ( k + 1 ) = X 1 σ ( k ) + μ P n H 11 σ Ψ 1 σ T G 11 σ P n , (73)
X 14 σ ( k + 1 ) = X 1 σ ( k ) + μ N 11 σ Ψ 1 σ T M 11 σ , (74)
X 21 σ ( k + 1 ) = X 2 σ ( k ) + μ P n A 12 σ T Ψ 1 σ B 12 σ T P n , (75)
X 22 σ ( k + 1 ) = X 2 σ ( k ) + μ C 12 σ T Ψ 1 σ D 12 σ T , (76)
X 23 σ ( k + 1 ) = X 2 σ ( k ) + μ P n H 12 σ Ψ 1 σ T G 12 σ P n , (77)
X 24 σ ( k + 1 ) = X 2 σ ( k ) + μ N 12 σ Ψ 1 σ T M 12 σ , (78)
X 15 σ ( k + 1 ) = X 1 σ ( k ) + μ P n A 21 σ T Ψ 2 σ B 21 σ T P n , (79)
X 16 σ ( k + 1 ) = X 1 σ ( k ) + μ C 21 σ T Ψ 2 σ D 21 σ T , (80)
X 17 σ ( k + 1 ) = X 1 σ ( k ) + μ P n H 21 σ Ψ 2 σ T G 21 σ P n , (81)
X 18 σ ( k + 1 ) = X 1 σ ( k ) + μ N 21 σ Ψ 2 σ T M 21 σ , (82)
X 25 σ ( k + 1 ) = X 2 σ ( k ) + μ P n A 22 σ T Ψ 2 σ B 22 σ T P n , (83)
X 26 σ ( k + 1 ) = X 2 σ ( k ) + μ C 22 σ T Ψ 2 σ D 22 σ T , (84)
X 27 σ ( k + 1 ) = X 2 σ ( k ) + μ P n H 22 σ Ψ 2 σ T G 22 σ P n , (85)
X 28 σ ( k + 1 ) = X 2 σ ( k ) + μ N 22 σ Ψ 2 σ T M 22 σ , (86)
X 1 σ ( k ) = 1 8 ∑ p = 1 8 X 1 p σ ( k ) , X 2 σ ( k ) = 1 8 ∑ q = 1 8 X 2 q σ ( k ) . (87)
Taking the vec-operator of Equations (71)-(86) respectively gives
col [ X 11 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( ( B 11 σ T P n ) T ⊗ P n A 11 σ T ) col [ Ψ 1 σ ] , (88)
col [ X 12 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( D 11 σ ⊗ C 11 σ T ) col [ Ψ 1 σ ] , (89)
col [ X 13 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( ( G 11 σ P n ) T ⊗ P n H 11 σ ) P ( 2 n ,2 n ) col [ Ψ 1 σ ] , (90)
col [ X 14 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( M 11 σ T ⊗ N 11 σ ) P ( 2 n ,2 n ) col [ Ψ 1 σ ] , (91)
col [ X 21 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( ( B 12 σ T P n ) T ⊗ P n A 12 σ T ) col [ Ψ 1 σ ] , (92)
col [ X 22 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( D 12 σ ⊗ C 12 σ T ) col [ Ψ 1 σ ] , (93)
col [ X 23 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( ( G 12 σ P n ) T ⊗ P n H 12 σ ) P ( 2 n ,2 n ) col [ Ψ 1 σ ] , (94)
col [ X 24 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( M 12 σ T ⊗ N 12 σ ) P ( 2 n ,2 n ) col [ Ψ 1 σ ] , (95)
col [ X 15 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( ( B 21 σ T P n ) T ⊗ P n A 21 σ T ) col [ Ψ 2 σ ] , (96)
col [ X 16 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( D 21 σ ⊗ C 21 σ T ) col [ Ψ 2 σ ] , (97)
col [ X 17 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( ( G 21 σ P n ) T ⊗ P n H 21 σ ) P ( 2 n ,2 n ) col [ Ψ 2 σ ] , (98)
col [ X 18 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ ( M 21 σ T ⊗ N 21 σ ) P ( 2 n ,2 n ) col [ Ψ 2 σ ] , (99)
col [ X 25 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( ( B 22 σ T P n ) T ⊗ P n A 22 σ T ) col [ Ψ 2 σ ] , (100)
col [ X 26 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( D 22 σ ⊗ C 22 σ T ) col [ Ψ 2 σ ] , (101)
col [ X 27 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( ( G 22 σ P n ) T ⊗ P n H 2 σ ) P ( 2 n ,2 n ) col [ Ψ 2 σ ] , (102)
col [ X 28 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ ( M 22 σ T ⊗ N 22 σ ) P ( 2 n ,2 n ) col [ Ψ 2 σ ] , (103)
col [ X 1 σ ( k ) ] = 1 8 ∑ p = 1 8 col [ X 1 p σ ( k ) ] , (104)
col [ X 2 σ ( k ) ] = 1 8 ∑ q = 1 8 col [ X 2 q σ ( k ) ] . (105)
Using notation in Equations (8) and Equations (104) and (105), the above equations can be equivalently written as
col [ X 1 σ ( k + 1 ) ] = col [ X 1 σ ( k ) ] + μ 8 [ Φ 11 σ T , Φ 21 σ T ] col [ Ψ 1 σ , Ψ 2 σ ] , (106)
col [ X 2 σ ( k + 1 ) ] = col [ X 2 σ ( k ) ] + μ 8 [ Φ 12 σ T , Φ 22 σ T ] col [ Ψ 1 σ , Ψ 2 σ ] . (107)
Combining these two equations gives
col [ X 1 σ ( k + 1 ) , X 2 σ ( k + 1 ) ] = col [ X 1 σ ( k ) , X 2 σ ( k ) ] + μ 8 [ Φ 11 σ T Φ 21 σ T Φ 12 σ T Φ 22 σ T ] col [ Ψ 1 σ , Ψ 2 σ ] = col [ X 1 σ ( k ) , X 2 σ ( k ) ] + μ 8 [ Φ 11 σ Φ 12 σ Φ 21 σ Φ 22 σ ] T col [ Ψ 1 σ , Ψ 2 σ ] = col [ X 1 σ ( k ) , X 2 σ ( k ) ] + μ 8 Φ σ T col [ Ψ 1 σ , Ψ 2 σ ] . (108)
According to notation in Equations (48) and (49) and using the conclusions in Equations (12) and (7), we have
col [ Ψ 1 σ , Ψ 2 σ ] = col [ ( F 1 − L 11 ( X 1 ( k ) ) − L 12 ( X 2 ( k ) ) ) σ , ( F 2 − L 21 ( X 1 ( k ) ) − L 22 ( X 2 ( k ) ) ) σ ] = col [ F 1 σ , F 2 σ ] − col [ ( L 11 ( X 1 ( k ) ) + L 12 ( X 2 ( k ) ) ) σ , ( L 21 ( X 1 ( k ) ) + L 22 ( X 2 ( k ) ) ) σ ]
= col [ F 1 σ , F 2 σ ] − col [ L 11 σ ( X 1 ( k ) ) + L 12 σ ( X 2 ( k ) ) , L 21 σ ( X 1 ( k ) ) + L 22 σ ( X 2 ( k ) ) ] = [ col [ F 1 σ ] col [ F 2 σ ] ] − [ col [ L 11 σ ( X 1 ( k ) ) + L 12 σ ( X 2 ( k ) ) ] col [ L 21 σ ( X 1 ( k ) ) + L 22 σ ( X 2 ( k ) ) ] ] = [ col [ F 1 σ ] col [ F 2 σ ] ] − [ Φ 11 σ col [ X 1 σ ( k ) ] + Φ 12 σ col [ X 2 σ ( k ) ] Φ 21 σ col [ X 1 σ ( k ) ] + Φ 22 σ col [ X 2 σ ( k ) ] ] = [ col [ F 1 σ ] col [ F 2 σ ] ] − [ Φ 11 σ Φ 12 σ Φ 21 σ Φ 22 σ ] [ col [ X 1 σ ( k ) ] col [ X 2 σ ( k ) ] ] (109)
Substituting Equation (109) into Equation (108) gives
col [ X 1 σ ( k + 1 ) , X 2 σ ( k + 1 ) ] = ( I 8 n 2 − μ 8 Φ σ T Φ σ ) col [ X 1 σ ( k ) , X 2 σ ( k ) ] + μ 8 Φ T col [ F 1 σ , F 2 σ ] . (110)
According to Lemma 4, if the algorithm in Equation (110) is convergent if and only if the convergence factor μ satisfies
0 < μ 8 < 2 λ m a x [ Φ σ T Φ σ ] . (111)
Namely, we have
0 < μ < 16 λ max 2 [ Φ σ ] .
The reminder of this theorem can be proved by referencing to Lemma 4. The proof is completed. □
Remark 1. Although some similar iterative algorithms have been established, in the preceding proof [
Remark 2. Although the sufficient and necessary condition of the convergence for algorithm in Equations (50)-(66) is given in Theorem 6, the Kronecker product and the real presentation of the coefficient matrices are involved in computing μ m a x , this results in the high dimensionality problem. To overcome this drawback, we give a sufficient condition for the suggested iterative algorithm.
Corollary 1. If the system of the complex coupled matrix Equations (5) has a unique solution pair X 1 ∗ and X 2 ∗ then the iterative algorithm in Equations (50)-(66) will converge to the solution pair X 1 ∗ and X 2 ∗ if
0 < μ ≤ 1 Σ = : μ 1 . (112)
Here
Σ : = ∑ i , j = 1 2 ( ‖ B i j ‖ 2 ‖ A i j ‖ 2 + ‖ D i j ‖ 2 ‖ C i j ‖ 2 + ‖ H i j ‖ 2 ‖ G i j ‖ 2 + ‖ N i j ‖ 2 ‖ M i j ‖ 2 ) .
Proof. Recall σ m a x [ A ] ≤ ‖ A ‖ and A ⊗ B = ‖ A ‖ ‖ B ‖ and refer to Lemma 3, we have
λ m a x 2 [ Φ σ ] = σ m a x 2 [ Φ σ ] ≤ ‖ Φ σ ‖ 2 = ‖ [ Φ 11 σ Φ 12 σ Φ 21 σ Φ 22 σ ] ‖ 2 = ‖ [ ‖ Φ 11 σ ‖ ‖ Φ 12 σ ‖ ‖ Φ 21 σ ‖ ‖ Φ 22 σ ‖ ] ‖ 2 = ( ∑ i , j = 1 2 ‖ Φ i j σ ‖ ) 2 ≤ 4 ∑ i , j = 1 2 ‖ Φ i j σ ‖ 2 . (113)
Noting the notation in Equation (8), we have
‖ Φ i j σ ‖ ≤ ‖ ( ( P n B i j σ ) T ⊗ A i j σ P n ) ‖ + ‖ ( D i j σ T ⊗ C i j σ ) ‖ + ‖ ( ( P n H i j σ ) T ⊗ G i j σ P n ) P ( 2 n ,2 n ) ‖ + ‖ ( N i j σ T ⊗ M i j σ ) P ( 2 n ,2 n ) ‖ = ‖ ( P n B i j σ ) T ‖ ‖ A i j σ P n ‖ + ‖ D i j σ T ‖ ‖ C i j σ ‖ + ‖ ( P n H i j σ ) T ‖ ‖ G i j σ P n ‖ + ‖ N i j σ T ‖ ‖ M i j σ ‖ = ‖ P n B i j σ ‖ ‖ A i j σ P n ‖ + ‖ D i j σ ‖ ‖ C i j σ ‖ + ‖ P n H i j σ ‖ ‖ G i j σ P n ‖ + ‖ N i j σ ‖ ‖ M i j σ ‖ = ‖ B i j σ ‖ ‖ A i j σ ‖ + ‖ D i j σ ‖ ‖ C i j σ ‖ + ‖ H i j σ ‖ ‖ G i j σ ‖ + ‖ N i j σ ‖ ‖ M i j σ ‖ = 4 ( ‖ B i j ‖ ‖ A i j ‖ + ‖ D i j ‖ ‖ C i j ‖ + ‖ H i j ‖ ‖ G i j ‖ + ‖ N i j ‖ ‖ M i j ‖ ) . (114)
Combining Equations (113) and (114) gives
λ m a x 2 [ Φ σ ] ≤ 16 ∑ i , j = 1 2 ( ‖ B i j ‖ ‖ A i j ‖ + ‖ D i j ‖ ‖ C i j ‖ + ‖ H i j ‖ ‖ G i j ‖ + ‖ N i j ‖ ‖ M i j ‖ ) 2 . (115)
Substituting Equation (115) into Equation (112) gives Equation (112). The proof is completed. □
In the practical work, condition
λ m a x 2 [ Φ σ ] = 16 ∑ i , j = 1 2 ( ‖ B i j ‖ ‖ A i j ‖ + ‖ D i j ‖ ‖ C i j ‖ + ‖ H i j ‖ ‖ G i j ‖ + ‖ N i j ‖ ‖ M i j ‖ ) 2
almost does not hold, so we use symbol “≤” to replace “<” in Corollary 1. Clearly, the Frobenius norm
∑ i , j = 1 2 ( ‖ B i j ‖ ‖ A i j ‖ + ‖ D i j ‖ ‖ C i j ‖ + ‖ H i j ‖ ‖ G i j ‖ + ‖ N i j ‖ ‖ M i j ‖ ) 2
is easier to compute then λ m a x 2 [ Φ σ ] in practical work.
In this section, we bring a numerical test to validate the efficacy of the related conclusions. Equation (5) includes some coupled complex matrix equations as its special cases. For convenience, we consider the following equation
{ A X B + C Y ¯ D = F 1 , G X T H + M Y H N = F 2 .
with
A = [ 19 + 11 i 1 4 + 9 i 19 + 13 i ] , B = [ 14 + 11 i 10 + 7 i 1 + 7 i 20 + 10 i ] ,
C = [ 17 + 20 i 0 5 + 9 i 20 + 20 i ] , D = [ 16 + 17 i 9 + 2 i 7 + 8 i 11 + 14 i ] ,
G = [ 14 + 20 i 5 + 5 i 5 + 3 i 13 + 16 i ] , H = [ 20 + 13 i 1 + i 2 + 8 i 15 + 16 i ] ,
M = [ 17 + 17 i 8 + 1 i 4 + 4 i 10 + 16 i ] , N = [ 16 + 10 i 5 + 7 i 6 + 9 i 11 + 11 i ] ,
F 1 = [ 1223 + 2480 i 2262 + 1209 i − 737 + 1962 i − 2060 + 1125 i ] ,
F 2 = [ 643 + 2484 i − 509 + 453 i 1132 + 43 i − 1455 − 538 i ] .
The unique solution pair of this system is
X * = [ 3 − i − 1 + i − 2 + 3 i ] , Y * = t [ 2 3 i 2 + i 1 − 2 i ] .
According to the suggested algorithm in Equations (50)-(66), we can construct the gradient-based algorithm for this system as follows:
X 1 ( k + 1 ) = X ( k ) + A H ( F 1 − A X ( k ) B − C Y ¯ ( k ) D ) B H , (116)
X 2 ( k + 1 ) = X ( k ) + H ¯ ( F 2 − G X T ( k ) H − M Y H ( k ) N ) T G ¯ , (117)
Y 1 ( k + 1 ) = Y ( k ) + C T ( F 1 − A X ( k ) B − C Y ¯ ( k ) D ) ¯ D T , (118)
Y 2 ( k + 1 ) = Y ( k ) + N ( F 2 − G X T ( k ) H − M Y H ( k ) N ) H M , (119)
X ( k ) = X 1 ( k ) + X 2 ( k ) 2 , Y ( k ) = Y 1 ( k ) + Y 2 ( k ) 2 . (120)
According to Corollary 1, a sufficient condition for the convergence factor μ is
0 < μ ≤ 4 ‖ A ‖ 2 ‖ B ‖ 2 + ‖ C ‖ 2 ‖ D ‖ 2 + ‖ G ‖ 2 ‖ H ‖ 2 + ‖ M ‖ 2 ‖ N ‖ 2 = 8.2144 × 10 − 7 = : μ 1 .
We take X ( 0 ) = Y ( 0 ) = 1 2 × 2 × 10 − 6 as the initial iterative value and use the iterative technique in Equations (116)-(120) to compute X ( k ) and Y ( k ) . The relative error with different convergence factors is shown in
δ : = ‖ X ( k ) − X ∗ ‖ 2 + ‖ Y ( k ) − Y ∗ ‖ 2 ‖ X ∗ ‖ 2 + ‖ Y ∗ ‖ 2
is the relative error.
From
λ m a x 2 [ Φ ] = 2.9622 × 10 6 < 4.8695 × 10 6 = ‖ B ‖ 2 2 ‖ A ‖ 2 2 + ‖ D ‖ 2 2 ‖ C ‖ 2 2 + ‖ G ‖ 2 2 ‖ H ‖ 2 2 + ‖ M ‖ 2 2 ‖ N ‖ 2 2 .
so the range of the convergence factor of Corollary 1 is conservative. Actually, 1.7 μ 1 plays the role of the best convergence factor in this numerical example.
This paper brings the gradient-based iterative technique for a coupled complex conjugate and transpose matrix equation. The convergence proof is offered. In the proof process, the necessary and sufficient conditions for the convergence factor are determined to guarantee the convergence of the algorithm. And the optimal choice of the convergence factor is settled theoretically. A numerical test is offered to validate the efficacy of the suggested algorithm. The iterative strategy used in this paper can be used in the system identification.
The authors declare no conflicts of interest regarding the publication of this paper.
Yin, H.C. and Zhang, H.M. (2021) Gradient-Based Iterative Algorithm for a Coupled Complex Conjugate and Transpose Matrix Equations. Advances in Linear Algebra & Matrix Theory, 11, 92-107. https://doi.org/10.4236/alamt.2021.113007