Coupled matrix equations have wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. When computing an additive decomposition of a generalized transfor- mation matrix [158], one naturally encounters the following generalized Sylvester
matrix equation
XA−DY=E
XC−BY =F , (1.35)
where X andY are the unknown matrices. In addition, the generalized Sylvester matrix equation (1.35) arises in computing the deflating subspace of descriptor lin- ear systems [159]. In stability analysis of linear jump systems with Markovian tran- sitions, the following two types of coupled Lyapunov matrix equations are often encountered [25, 185]
ATiPi+PiAi+Qi+ N
j=1
pijPj=0 ,Qi>0,i∈I[1,N], (1.36)
and
Pi=ATi
⎛
⎝N
j=1
pijPj
⎞
⎠Ai+Qi,Qi>0,i∈I[1,N], (1.37)
where Pj,j ∈ I[1,N], are the matrices to be determined. A more general coupled matrix equation has the following form
Ai1X1Bi1+Ai2X2Bi2+ ã ã ã +AipXpBip=Ei,i∈I[1,N], (1.38) whereXj,j∈I[1,p], are the matrices to be determined.
Due to their broad applications, coupled matrix equations have attracted con- siderable attention from many researchers. In [236], the following coupled matrix equation was investigated:
A1XB1 =C1 A2XB2 =C2 .
It was shown in [236] that this coupled equation is solvable if and only if the following conditions hold:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩ rank
⎡
⎣A1C1 0 A2 0 −C2
0 B1 B2
⎤
⎦=rank A1
A2
+rank B1 B2
,
rank A1C1
=rankA1, rank
A2C2
=rankA2,
rank C1
B1
=rankB1, rank
C2
B2
=rankB2.
It was pointed out in [43] that the existence of a positive definite solution to the discrete-time Markovian jump Lyapunov (DMJL) matrix equation (1.37) is related to the spectral radius of an augmented matrix required to be less than one. WhenA,B, C, andDare square matrices, it was shown in [41] that the matrix equation (1.35) has a unique solution if and only if the spectra of the pencils(A−sC)and(D−sB)have an empty intersection. This conclusion implies a dual relation between the matrix equations (1.35) and (1.15). The existence condition of a solution to the equation (1.35) with squareA,B,C, andDwas given in [240].
A direct method for solving these kinds of coupled matrix equations is to convert them into matrix-vector equations by using the Kronecker product. With this idea, a simple and elegant explicit solution to the DMJL matrix equation (1.37) was given in [43] in terms of matrix inverses. However, computational difficulties arise when the dimensions of the coefficient matrices are high. To eliminate the high dimension- ality problem, one can first transform coefficient matrices into some special forms, and then solve some matrix equations to be more easily dealt with. Such a kind of methods belongs to the so-called transformation approaches, and has been widely adopted. For example, the generalized Schur method by applying the QZ algorithm was developed to solve the generalized Sylvester matrix equation (1.35) in [159]. In
[177], based on the projection theorem in Hilbert space, by making use of the gener- alized singular value decompositions and the canonical correlation decompositions, an analytical expression of the least-squares solution to the simple coupled matrix equation(AXB,GXH)=(C,D)was established.
Another kind of efficient approaches to solve coupled matrix equations is the type of iterative algorithms. A parallel iterative scheme for solving the DMJL equation (1.37) was proposed in [26]. Under the condition that the corresponding Markovian jump system is stochastically stable, this method was proven to be convergent to the exact solution if the initial condition chosen is zero. In [237], the restriction of zero initial conditions in [26] was removed, and a new iterative method was provided by using the solutions of standard discrete-time Lyapunov equations as its intermediate steps. In [248], two new iterative algorithms were developed for solving the coupled Lyapunov matrix equation (1.37) by using the latest estimation. The idea in [26] was also extended to solve the continuous-time Markovian jump Lyapunov (CMJL) matrix equation (1.36) in [25]. By using the concept of latest estimation proposed in [248], an implicit sequential algorithm was proposed in [201] to solve the CMJL matrix equation (1.36) based on the algorithm in [26]. Besides, implicit iterative algorithms with some tunable parameters were developed in [255] to solve the coupled Lyapunov matrix equation (1.36). A significant feature of the algorithms in [255] is that the iterative sequences are updated by using not only the information in the last step, but also the information in the current step and the previous steps.
The aforementioned so-called parallel scheme is only valid for some coupled matrix equations with special structure, and is not applicable to general coupled matrix equations. For some general coupled matrix equations, the hierarchical- identification-principle-based technique was developed by Ding and his co-authors.
The basic idea of such an algorithm is to regard the unknown matrices to be found as the system parameters to be identified. In [56], a gradient-based iterative algo- rithm was proposed for (1.38) with unique solutions by the hierarchical identification principle. In [55], a least-squares iterative algorithm was proposed to solve the gen- eralized Sylvester matrix equation (1.35) based on the hierarchical identification principle [53, 54]. This idea was also used in [55] to solve the general coupled matrix equation (1.38) withp =N. In order to obtain the solution, the concept of the block-matrix inner product was introduced in [55].
Another approach for constructing iterative algorithms is to adopt the concept of optimization. With this idea, a gradient-based algorithm was developed in [309] to solve the coupled Lyapunov matrix equation (1.37). This approach has been gener- alized to some more complicated cases. Gradient-based iterations were constructed in [306] to solve the general coupled matrix equation (1.38). A significant character- istic of the method in [306] is that a necessary and sufficient condition guaranteeing the convergence of the algorithm can be explicitly obtained. Moreover, the algo- rithm in [306] removed the restriction ofp =N. In [311] gradient-based iterative algorithms were also proposed to obtain the weighted least-squares solution to the general coupled Sylvester matrix equation (1.38). Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms were also derived.
The aforementioned iterative algorithms are only applicable to the case where the involved matrix equation has a unique solution. For a coupled matrix equation with infinitely many solutions, it is suitable to adopt finite iterative methods. This kind of approaches is very similar to the conjugate gradient approach for solving ordinary linear matrix-vector equations. The basic idea is to construct a sequence of orthogonal residuals in an appropriate inner product space. With this idea, iterative algorithms were proposed for solving the DMJL matrix equation (1.37) in [229]
and CMJL matrix equation (1.36) in [279]. By using the inner product as a tool, it was proven that the proposed algorithm can obtain the exact solution within finite iteration steps. Moreover, the algorithm can be implemented without transformation of original coefficient matrices into any canonical forms. In [49], a finite iterative algorithm was proposed to obtain the reflexive solution to the generalized Sylvester matrix equation (1.35). The method in [49] was generalized in [51] to obtain the bisymmetric solution of the matrix equation (1.38) with p = N = 2. In [135], the matrix form of the bi-conjugate gradient stabilized (Bi-CGSTAB) method was established for solving a kind of coupled matrix equations with two unknown matrices by employing Kronecker products and vectorization operators.