=Paper=
{{Paper
|id=Vol-3687/Short_7
|storemode=property
|title=Algorithm for Finding a Positive Definite Solution to the Sylvester Matrix Equation
|pdfUrl=https://ceur-ws.org/Vol-3687/Short_7.pdf
|volume=Vol-3687
|authors=Oleksii Bychkov,Ganna Marzafey,Georgy Dimitrov
|dblpUrl=https://dblp.org/rec/conf/dsmsi/BychkovMD23
}}
==Algorithm for Finding a Positive Definite Solution to the Sylvester Matrix Equation==
<pdf width="1500px">https://ceur-ws.org/Vol-3687/Short_7.pdf</pdf>
<pre>
                         Algorithm for Finding a Positive Definite Solution to the
                         Sylvester Matrix Equation
                         Oleksii Bychkov 1, Ganna Marzafey 2 and George Dimitrov 3
                         1
                          Taras Shevcenko National University of Kyjv, Volodymyrska str. 60, Kyjv, Ukraine, 01601
                         2
                          University of Library Studies and Information Technologies, Blvd. “Tsarigradsko Shose” 119, 7-Kilometar,
                         Sofia, Bulgaria, 1784

                                         Abstract
                                         The article examines the problem of constructing an algorithm for finding positive definite
                                         matrices that are the solution to Sylvester's three-term matrix equation. The problem is that,
                                         unlike the Lyapunov equation, such a condition cannot be written in terms of eigenvalues. The
                                         condition for the existence of a solution to the Sylvester equation is based on the principle of
                                         contraction mappings. The article also proposes an iterative procedure and algorithm for
                                         finding a solution.
                                         Keywords1
                                         algorithm, stability, iteration procedure, Silvestr, solution

                         1. Introduction

                             The Sylvester matrix equation 𝐴𝑋 + 𝑋𝐵 = 𝐶, also sometimes called the continuous Sylvester
                         equation, and the Stein matrix equation 𝑋 + 𝐴𝑋𝐵 = 𝐶, in turn, sometimes called the discrete
                         Sylvester equation, as well as their special cases - the Lyapunov equations 𝐴𝑇 𝑋 + 𝑋𝐴 = 𝐶 and 𝑋 −
                          𝐴𝑋𝐴𝑇 = 𝐶, are well-studied and frequently encountered (for example, in the theory of differential
                         equations) classes of matrix equations [1-6]. The conditions for the unique solvability of these equations
                         have long been known; there are numerical algorithms for solving them, for example, the Bartels-
                         Stewart and Golub-Nash-Van Loan algorithms.
                             An analogue of Sylvester's equation 𝐴𝑋 + 𝑋𝐵 = 𝐶 began to attract the attention of researchers
                         relatively recently [7-9]. Although this equation is superficially very similar to Sylvester's equation,
                         their natures are profoundly different. Let's give a simple example to illustrate this difference. If all
                         matrices are square and 𝐴 = 𝐵, then Sylvester’s equation has a unique solution 𝑋 - for any right-hand
                         side 𝐶. For the same coefficients 𝐴 and 𝐵, the equation 𝐴𝑋 + 𝑋𝐵 = 𝐶 has a solution only if the matrix
                         𝐶 is symmetric. If this condition is met and 𝑋 satisfies this equation, then 𝑋 + 𝐾, where 𝐾 is an
                         arbitrary skew-symmetric matrix, is also a solution.
                                         𝑇
                             Equation 𝐴 𝑋 + 𝑋𝐴 = 𝐶 , as well as the equations 𝐴𝑋 + 𝑋𝐵 = 𝐶 we will generally call two
                         member equations of Sylvester type. The relevance of studying this kind of equations is beyond doubt.
                         Let us give several examples showing why the study of equations of Sylvester type is justified. Equation
                         𝐴𝑋 + 𝑋𝐵 = 𝐶 was first encountered by us in article [10-12], where it was studied under the additional
                         assumption 𝐶 = 𝐶 𝑇 .
                             Solvability conditions and a description of the general solution were given in terms of generalized
                         inverses for matrices 𝐴 and 𝐵 and their associated projectors. These conditions are not entirely


                         Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine
                         EMAIL: oleksiibychkov@knu.ua (O. Bychkov); annamartsafei@knu.ua (G, Marzafey); geo.p.dimitrov@gmail.com (G. Dimitrov)
                         ORCID: 0000-0002-9378-9535 (O. Bychkov); 0000-0001-5064-3168 (G. Dimitrov)
                                      ©️ 2023 Copyright for this paper by its authors.
                                      Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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constructive and are difficult to verify. Homogeneous equation 𝐴𝑋 + 𝑋𝐵 = 𝐶 was studied in article
[2] in the special case 𝐵 = 𝐴. The authors were motivated by the fact that the set 𝐴𝑋 + 𝑋𝐵 ∈ 𝐶 𝑛×𝑛
is the tangent space (calculated at point 𝐴) of the orbit of matrix 𝐴 under the action of congruences. The
codimension of this orbit is exactly the dimension of the space of solutions to the equation 𝐴𝑋 + 𝐴𝑋 =
 0. The main result of work [2,3] was the establishment of the canonical structure of matrices in general
position with respect to congruences. In a similar way, the same authors in [4] studied the equation
𝐴𝑋 + 𝐴𝑋 = 0. In the publication [4] the equations 𝐴𝑋 + 𝑋𝐵 = 𝐶 arise when constructing an
algorithm for palindromic eigenvalue problems 𝐴𝑥 = 𝜆𝐴𝑇 𝑥 .
    In the process of reducing matrix A to antitriangular form, the need arises to solve these equations
numerically. In article [5], the conditions for unique solvability and a numerical algorithm for solving
the equation 𝐴𝑋 + 𝑋𝐵 = 𝐶 are formulated.
    By analogy with equations of Sylvester type, equations of Stein type will be called equations 𝑋 +
 𝐴𝑋𝐵 = 𝐶. The first of them was partially studied in publication [6]. The question naturally arises
about the completion of this study. The topic of solvability of the other two equations, on the contrary,
is explored exhaustively in this publication.
    No less interesting is the problem of obtaining sufficient solvability conditions for the Sylvester
matrix equation
                                           𝐴𝑇 𝐻 + 𝐻𝐴 + 𝐵𝑇 𝐻𝐵 = −𝐶,
where 𝐶 is a positive definite matrix. The need to solve such equations is emphasized in [13-14].
    It is noted in [9] that there are now several approaches to solving the Sylvester equation. The first is
to reduce the matrix equation to a vector (linear algebraic equation of increased dimension) and then
the condition for the solvability of this equation is expressed through the non-degeneracy of the
corresponding matrix. The second approach uses the small parameter method. It is also possible to
obtain a spectral sparsity criterion for an equation with mutually commutable matrices. An analytical
solution of this equation is possible only for the case of the two-term Sylvester equation.
    At the same time, the question of the existence of a positive definite solution in the general case
remains open. The procedure for finding this solution is also of interest for another investigations [15-
16]. The purpose of the article is to obtain sufficient conditions for the existence of a solution to the
Sylvester matrix equation on a set of positive definite matrices. The article will also present an iterative
procedure and algorithm for finding these solutions.

2. Main result

   The approach is based on a modification of the contraction mapping method. If on a complete
metric space 𝑀 the following operator is specified 𝐹[𝑥], 𝑥 ∈ 𝑀, that maps points 𝑥 ∈ 𝑀to points of the
same space 𝐹[𝑥] ∈ 𝑀 and satisfies the contraction condition 𝜌(𝐹[𝑥], 𝐹[𝑦]) ≤ 𝛼𝜌(𝑥, 𝑦), 0 < 𝛼 < 1,
where 𝜌(𝑥, 𝑦) is the metric of the space M, then the operator equation
                                                 𝑥 = 𝐹[𝑥]
has a unique solution 𝑥 ∗ ∈ 𝑀, and it can be found by the method of successive iterations
                            𝑥 = lim 𝑥𝑛 , 𝑥𝑛 = 𝐹[𝑥𝑛−1 ], 𝑛 = 1,2, … , 𝑥0 = 𝑥 0 .
                                 𝑛→∞

   Let 𝑥0 - be an arbitrary point in space, H is some complete metric space with metric 𝜌(𝐻1 , 𝐻2 ). Let
us fix the following operators:
                                              𝐹, 𝐺: 𝐻 → 𝐻.
   Consider the operator equation
                                               𝐹[𝐻] = 𝐺[𝐻]                                              (1)



                                                                                                        138
   Let us show that to solve this equation we can use an iterative procedure based on the following
implicit scheme
                           𝐹[𝐻𝑠+1 ] = 𝐺[𝐻𝑠 ], 𝐻0 = 𝐻0 , 𝑠 = 1,2, . . , ; 𝐻0 ∈H                         (2)
  Definition. We call method (2) converging to the solution of equation (1) if the sequence {𝑦𝑠 }, 𝑦𝑠 =
𝐹[𝐻𝑠 ], at 𝑠 → ∞ converges. Let us prove a theorem whose conditions ensure the convergence of
method (2) to the solution of equation (1).
  Let us assume that the solution to equation (1) lies in the set 𝐻(𝐴, 𝑟):
                                    𝐻 (𝐴, 𝑟) = {𝐻 ∈ 𝐻: 𝜌 (𝐹[𝐻], 𝐹[𝐴])⟨𝑟}.
    Theorem 1. Let us fix the set 𝐻 (𝐴, 𝑟), i.e. the values of 𝑟 and 𝐴 are determined. If for arbitrary
𝐻1 , 𝐻2 ∈ 𝐻(𝐴, 𝑟) operators 𝐹, 𝐺: 𝐻 → 𝐻 satisfy the contraction condition
                               𝜌(𝐺[𝐻1 ], 𝐺[𝐻2 ]) < 𝑞 𝜌(𝐹[𝐻1 ], 𝐹[𝐻2 ]), 0 < 𝑞 < 1,                    (3)
   and
                                         𝜌(𝐺[𝐴], 𝐹[𝐴]) < (1 − 𝑞)𝑟 ,                                    (4)
   then operator equation (1) has a unique solution in the set 𝐻(𝑎, 𝑟) and the sequence {𝑦𝑠 }, 𝑦𝑠 = 𝐹[𝐻𝑠 ]
constructed according to scheme (2) converges to 𝑦 ∗ = 𝐹[𝐻∗ ] for any starting point 𝐻0 𝜖𝐻 (𝐴, 𝑟). For
the error of method (2), the following estimate is valid:
                                                    𝑞𝑘
                               𝜌(𝐹[𝐻𝑘 ], 𝐹[𝐻 ∗ ]) < 1−𝑞 𝜌(𝐺[𝐻0 ], 𝐹[𝐻0 ]).                            (5)

   Proof. Let 𝐻0 - be an arbitraryelement of H(A, r) . Let us prove that the sequence 𝐻𝑘 built according
to scheme (2) will not leave the set H(A, r). By condition 𝐻0 ∈ 𝐻 (𝐴, 𝑟). Let us assume that 𝐻𝑘 ∈
𝐻 (𝐴, 𝑟)for some fixed k. Let us show that then 𝐻𝑘+1 ∈ 𝐻 (𝐴, 𝑟). Consider the equality:
                                              𝐹[𝐻𝑠+1 ] = 𝐺[𝐻𝑠 ]
   Let us subtract the 𝐹[𝐴] from both sides of the above mentioned equation, we get
               𝐹[𝐻𝑘+1 ] − 𝐹[𝐴] = 𝐺[𝐻𝑘 ] − 𝐹[𝐴] = (𝐺[𝐻𝑘 ] − 𝐺[𝐴]) + (𝐺[𝐴] − 𝐹[𝐴])
   Then the inequality holds
                          𝜌(𝐹[𝐻𝑘+1 ], 𝐹[𝐴]) < 𝜌(𝐺[𝐻𝑘 ], 𝐺[𝐴]) + 𝜌(𝐺[𝐴], 𝐹[𝐴]).
   Using the induction hypothesis, we have
                                 𝜌(𝐺[𝐻𝑘 ], 𝐺[𝐴]) < 𝑞𝜌(𝐹[𝐻𝑘 ], 𝐹[𝐴]) < qr
   and since by the conditions of the theorem 𝜌(𝐺[𝐴], 𝐹[𝐴]) < (1 − 𝑞)𝑟 we get that
                                    𝜌(𝐹[𝐻𝑘+1 ], 𝐹[𝐴]) < qr + (1 − 𝑞)𝑟 = 𝑟.
    Thus 𝐻𝑘+1 ∈ 𝐻(𝐴, 𝑟).
    Now we will show that the following sequence {𝑦𝑠 }, 𝑦𝑠 = 𝐹[𝐻𝑠 ] is Cauchy. Let's consider the
difference 𝐹[𝐻𝑘+1 ] − 𝐹[𝐻𝑘 ]:
                                       𝐹[𝐻𝑘+1 ] − 𝐹[𝐻𝑘 ] = 𝐺[𝐻𝑘 ] − 𝐺[𝐻𝑘−1 ]
   Because the 𝐻𝑘 ∈ 𝐻(𝐴, 𝑟), then using condition (3) we obtain
                    𝜌(𝐹[𝐻𝑘+1 ], 𝐹[𝐻𝑘 ]) = 𝜌(𝐺[𝐻𝑘 ], 𝐺[𝐻𝑘−1 ]) < 𝑞𝜌(𝐹[𝐻𝑘 ], 𝐹[𝐻𝑘−1 ])
   and therefore
                         𝜌(𝐹[𝐻𝑘+1 ], 𝐹[𝐻𝑘 ]) < 𝑞𝑘 𝜌(𝐹[𝐻1 ], 𝐹[𝐻0 ])                                  (6)


   Let 𝑝 ∈ 𝑁. Then it's fair
                            𝐹[𝐻𝑘+𝑝 ] − 𝐹[𝐻𝑘 ] = ∑𝑝𝑗=1(𝐹[𝐻𝑘+𝑗 ] − 𝐹[𝐻𝑘+𝑗−1 ]).
   And according to (6) we get:


                                                                                                      139
                                                       𝑞𝑘                                           (7)
                            𝜌(𝐹[𝐻𝑘+𝑝 ], 𝐹[𝐻𝑘 ]) < 1−𝑞 𝜌(𝐹[𝐻1 ], 𝐹[𝐻0 ]).
                   𝑞𝑘
   Because lim 1−𝑞 = 0 and does not depend on 𝑝, then the sequence {𝑦𝑠 } - Cauchy. Therefore there
             𝑘→¥
is a limit
                                      lim 𝑦𝑠 = 𝑦 ∗ , 𝑦 ∗ = 𝐹[𝐻 ∗ ], 𝐻 ∗ ∈ 𝐻 (𝐴, 𝑟)
                                      𝑘→∞


   Let us pass to the limit in (2) at 𝑘 → ∞, then we get that 𝐹[𝐻∗ ] = 𝐺[𝐻 ∗ ]. Hence 𝐻 ∗ - is a solution
of equation (1). Let us prove the uniqueness of this solution. Let 𝐻∗ is a some another solution (1) and
𝐻∗ ∈ 𝐻(𝐴, 𝑟). Then
                                        𝐹[𝐻∗ ] − 𝐹[𝐻 ∗ ] = 𝐺[𝐻∗ ] − 𝐺[𝐻∗ ]
and
                                         𝜌(𝐹[𝐻∗ ], 𝐹[𝐻 ∗ ]) < 𝑞𝜌(𝐹[𝐻∗ ], 𝐹[𝐻 ∗ ]).
   Because 0 < 𝑞 < 1, That 𝐻∗ = 𝐻 ∗ .
   Let us prove the estimate of the error of method (2). Let k be fixed and 𝑝 → ∞. Then from (7) we
obtain
                                                  𝑞𝑘
                             𝜌(𝐹[𝐻𝑘 ], 𝐹[𝐻∗ ]) < 1−𝑞 𝜌(𝐹[𝐻1 ], 𝐹[𝐻0 ]), 𝑘 = 1,2, 𝐾.
   Therefore the theorem is proven.

2.1.     Existance of solution

   Denote as H the space of symmetric matrices with the metric -𝜌(𝐻1 , 𝐻2 ) = |𝐻1 − 𝐻2 |.
   Let us write the Lagrange formula for the operators F and G. We have:
                   𝐹[𝐻1 ] − 𝐹[𝐻2 ] = 𝐻2 (𝑥)(𝐻1 − 𝐻2 ), 𝐺[𝐻1 ] − 𝐺[𝐻2 ] = 𝐻1 (𝑥)(𝐻1 − 𝐻2 ),
   Where 𝐻1 (.), 𝐻2 (.) are the Gâteaux derivatives of the operators 𝐹 and 𝐺 at some midpoint.
   Assume that there is an inverse operator 𝐻2−1 . Then we get:
                              𝐺[𝐻1 ] − 𝐺[𝐻2 ] = 𝐻1 (𝑥)𝐻2−1 (𝑥)(𝐹[𝐻1 ] − 𝐹[𝐻2 ]).
   Thus we have
                        | 𝐺[𝐻1 ] − 𝐺[𝐻2 ]| ≤ |𝐻1 (𝑥)𝐻2−1 (𝑥)||𝐹[𝐻1 ] − 𝐹[𝐻2 ]|                      (8)
   It follows that condition (3) of Theorem 1 will be satisfied if
                                      |𝐻1 (𝑥)𝐻2−1 (𝑥)| ≤ 𝑞 < 1.                                     (9)
   This condition is not always convenient. Using Theorem 1 and inequality (9), we obtain the
conditions for the solvability of the Sylvester matrix equation.
   Let us define the operators F and G as follows:
                                   𝐹[𝐻]=-𝐴𝑇 𝐻 − HA, 𝐺[𝐻] = 𝐶 + 𝐵𝑇 HB,
   where 𝐴, 𝐵 are some matrices, 𝐶 is a positive definite matrix. Then the Sylvester matrix equation
can be rewritten as:
                                       𝐹[𝐻] = 𝐺[𝐻].                                            (10)
   Let us obtain constructive solvability conditions for this equation.
   Lemma 1. A necessary condition for the solvability of the Sylvester matrix equation on the set of
positive definite matrices is that the matrix 𝐴 is Hurwitz.
   Proof. Let 𝐻0 is a positive definite solution to Sylvester's equation. Consider the expression
                                        −𝐴𝑇 𝐻0 − 𝐻0 𝐴 = 𝐶 + 𝐵𝑇 𝐻0 𝐵.
   Let's denote 𝐶1 = 𝐶 + 𝐵𝑇 𝐻0 𝐵. Because 𝐶 and 𝐻0 are positive definite, then 𝐶1 will be a positive
definite. Then

                                                                                                     140
                                           −𝐴𝑇 𝐻0 − 𝐻0 𝐴 = 𝐶1
   is a matrix Lyapunov equation and since 𝐻0 is his solution, then 𝐴 is a Hurwitz matrix.

2.2.    Iteration procedure

   Lemma 2. Let matrices 𝐴, 𝐵 satisfy the following conditions:
                                      1) 𝐴- is Hurwitz matrix,                                         (11)
                               2)|(𝐵𝑇 ×B)(−𝐴𝑇 ×I − I×A)−1 | ≤ 𝑞,                                       (12)

   where × denotes the Kronecker product, then there is a unique positive definite solution to the
Sylvester matrix equation and it can be found using the iterative procedure
                                       −𝐴𝑇 𝐻𝑘+1 − 𝐻𝑘+1 𝐴 = 𝐶 + 𝐵𝑇 𝐻𝑘 𝐵,
where 𝐶 is an arbitrary positive definite matrix.
  Proof. Let us apply inequality (9) to the operators defining the Sylvester equation. For operator 𝐺
we have:
                                                  𝛨1 (𝜉 ) = 𝐵𝑇 ×B.
   Since 𝐴 is Hurwitz, then the operator 𝐻2−1 (𝜉 ) exists and can be written
                                          𝛨2−1 (𝜉 ) = (−𝐴𝑇 xI − IxA)−1 .
   Consequently, we find that inequality (12) ensures that condition (3) of Theorem 1 is satisfied.
Let us take as the working set the entire space of positive definite matrices, then 𝑟=∞. Thus, we obtain
conditions for the existence and uniqueness of a solution to the Sylvester matrix equation.
   Let us show that the solution obtained by the iterative procedure
                                        −𝐴𝑇 𝐻𝑘+1 − 𝐻𝑘+1 𝐴 = 𝐶 + 𝐵𝑇 𝐻𝑘 𝐵
will be a positive definite matrix. Let us prove this by mathematical induction. Matrix 𝐻0 positive
definite - due to the choice of the starting point. Let 𝐻𝑘 is a positive definite matrix. Let's show positive
definiteness 𝐻𝑘+1 .
   Consider the equation
                                              −𝐴𝑇 𝐻𝑘+1 − 𝐻𝑘+1 𝐴 = 𝐶𝑘 ,
                     𝑇
where 𝐶𝑘 = 𝐶 + 𝐵 𝐻𝑘 𝐵,𝑘 = 1,..., 𝑛,...
   Since 𝐴 is Hurwitz and 𝐶𝑘 - is always positive definite, then𝐻𝑘+1 , as a solution to the Lyapunov
equation will be a positive definite matrix. The lemma is proven.

2.3.     Application examples

        Let us consider examples of the fulfillment of the conditions of Lemma 2. Let in the first
example
                                      3 5 0                0.1 0.5 0
                                 𝐴 = (2 1 0) 𝐵 = ( 4 0.3 0.9)
                                      0 3 5                0.1 0 0.5
where matrix 𝐴 is Hurwitz. We substitute the values of matrices 𝐴 and 𝐵 into condition (12), carry out
calculations and find that all elements in the resulting matrix are less than 1 in absolute value, which
indicates that the condition is met.
   In the second example
                                      1 2 0                0.1 0.2 0.1
                                 𝐴 = (3 4 0) 𝐵 = (0.3 0.1 0.5)
                                      0 1 2                0.2 0.4 0.1


                                                                                                        141
  we carry out similar calculations and obtain the fulfillment of conditions (11) and (12).
  Let us give an example of non-fulfillment of the conditions of Lemma 2. Let the values of the
matrices be as follows:
                                      1 2 0                   1 2 3
                                 𝐴 = (3 4 0)           𝐵 = (3 4 5)
                                      0 1 2                   2 4 1

   In this case, when calculating condition (12), we obtain a matrix where some elements have absolute
values greater than 1, and this indicates that the condition is not met.

2.4.    Algoritm for solving Silvest’r equation

   Let us consider an algorithm for finding positive definite solutions. For finding a solution of the
Sylvester equation, we apply the following algorithm:
   1. Check the Hurwitz property of matrix 𝐴.
   2. Check the condition |(𝐵𝑇 ×B)(−𝐴𝑇 ×I − I×A)−1 | ≤ 𝑞.
   3. If 1.2 are true, then a solution exists and perform step 4. Otherwise, step 9.
   4. We let 𝑘 = 0, 𝐻0 = 𝐼, 𝐼 − identity matrix.
   5. At the kth step we calculate 𝐶1 = 𝐶 + 𝐵𝑇 𝐻𝑘 𝐵.
   6. Solve the Lyapunov equation −𝐴𝑇 𝐻𝑘+1 − 𝐻𝑘+1 𝐴 = 𝐶1 .
   7. Check the condition for ending the iteration procedure. If it is not fulfilled, then 𝑘 = 𝑘 + 1 and
go to step 5, otherwise go to step 8.
   8. Resulting matrix 𝐻𝑘 - is a solution to the Sylvester matrix solution.
   9. The end.

3. Acknowledgements

   We thanks professor Khusainov Denys and Erasmus+ for supporting our research.

4. Conclusions and discussion of results

    The article studies the solvability of Sylvester's three-term matrix equation. Using the principle of
contraction mappings, a sufficient condition for the existence of a positive definite solution is obtained.
It should be noted that such conditions have not been published before. In other works authors solve
similar equations by expanding the equation into a system of linear algebraic equations with constant
coefficients and then solving it, for example, using the Gauss method. This approach does not allow us
to obtain the condition for the existence of the necessary solution. In this article, in addition to the
sufficient condition for the existence of a positive definite solution to the three-term matrix Sylvester
equation, an algorithm for finding it is proposed and the convergence of this algorithm is proven.
    The method presented in this article will allow us to more effectively solve problems of stability and
controllability, which are presented in [17-22]. The solvability condition for Sylvester's three-term
matrix equation is easy to verify. It is constructive. The proposed algorithm effectively finds a
symmetric positive definite solution.

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