The Simultaneous Reduction of Matrices to the Block-Triangular Form

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1 Physics Journal Vol., No., 05, htt:// The Simultaneous Reduction of Matrices to the Bloc-Triangular Form Yuri N. Bazileich * Deartment of Alied Mathematics, Prydniros State Academy of Ciil Engineering and Architecture, Dniroetros', Uraine Abstract The solution of the roblem of seeral n n matrices reduction to the same uer bloc-triangular form by a similarity transformation with the greatest ossible number of blocs on the main diagonal is gien. In addition to the well-nown "method of commutatie matrix" a new "method of inariant subsace" is used. Keywords Matrix, Bloc-Triangular Form, Similarity Transformation, the Centralizer, Algebra oer the Field, Radical Ideal Receied: July 0, 05 / Acceted: July 5, 05 / Published online: August 8, 05 The Authors. Published by American Institute of Science. This Oen Access article is under the CC BY-NC license. htt://creatiecommons.org/licenses/by-nc/4.0/. Introduction At first, let us consider the case when the number of matrices d =.... l 0... l =, =, ll There are two square matrices B and B oer the field C of comlex numbers. We need to find the similarity transformation () reducing both matrices to the equal bloctriangular form. Here B ɶ i is a bloc of matrix B located in the i-th column and -th row of this artitioned matrix. The diagonal blocs must be square submatrices. It is necessary that the number l of diagonal blocs to be the maximum ossible. The idea of the method was ublished in the author's monograh []. Corresonding comutational algorithms and results of calculations on handling of alied roblems are resented in aers [, 3]. In this aer a detailed theoretical () basis of the deeloed methods is gien. There exists a solution for the case of only one matrix. One matrix can be reduced to its Jordan form. There is a famous unsoled roblem to create a canonical form for a air of matrices. This roblem and the equialent roblems are called wild roblems [4].. The General Calculation Scheme We use the "method of commutatie matrix" and the "method of inariant subsace." The first one allows you to find the similarity transformation, reducing both matrices to a blocdiagonal form with two (at least) blocs on the main diagonal, or to determine that such reduction is imossible for these matrices. The other method is intended to reduce such two matrices that are not reduced to the bloc-diagonal form, to a bloc-triangular form, or to determine that they cannot be reduced to a "strict" bloc-triangular form either. There is also used a method of oercoming the secial case (see Section 6). * Corresonding author address: bazileich@yandex.ru

2 55 Yuri N. Bazileich: The Simultaneous Reduction of Matrices to the Bloc-Triangular Form This aroach is consistently alied firstly to the initial air of matrices, then to airs of blocs that aear on the main diagonal. The rocess continues until we obtain the airs of blocs that are irreducible to the same bloc-triangular form. From the uniqueness theorem it follows that this aroach gies the solution of the roblem. 3. The Method of Commutatie Matrix Possibility of alication of commutatie matrix for decouling of system of equations was described in the textboos on quantum mechanics (for examle, see the boo by Fermi [5]). The method of commutatie matrix as such was roosed simultaneously by A.K. Loatin [6] and E.D. Yauboich [7]. Let us consider Λ ( B ) set of all matrices that are commutatie with matrices B, B. This set is an algebra oer the field C of comlex numbers. Λ( ) is called a centralizer of matrices {B }. Theorem. Let matrices A and X be commutatie: AХ = ХA, matrix X has the bloc-diagonal form of B X = diagх, where the sectra of the blocsх are mutually disoint. Then the matrix А also has the bloc-diagonal form А = diagа. This theorem is gien in [8] Ch. VIII, Theorem 3. See also [].5. Corollary. Let a matrix X Λ ( B ) exist, haing at least two different eigenalues. Let the column of matrix S be the ectors of canonical basis of matrix X. Then similarity transformation reduces both matrices to the bloc-diagonal form with two (at least) blocs on the main diagonal. Proof. Proerty of matrices commutation is resered under the similarity transformation. Indeed: where AB BA S ABS S BAS S ASS BS = = = S AS AB ɶɶ = BA ɶ ɶ, ɶ, ɶ, S is the non-singular matrix. A S = AS B Let the matrix X hae at least two different eigenalues and commute with matrices B, B and let the columns of the matrix S be the ectors of the canonical basis of the matrix X. In this case the transformation Xɶ = S XS reduces matrix X X 0 to its Jordan form. Therefore, Xɶ = = diag( X, X) 0 X, where X is a Jordan bloc corresonding to the first eigenalue of the matrix X, and X to the second and the subsequent (if any) eigenalues. Both blocs are not emty and they hae no common eigenalues. The matrices commute with the matrixx ɶ. From Theorem it follows that they hae the bloc-diagonal form. Theorem. If matrices B are reduced to the bloc-diagonal form with two (at least) blocs on the main diagonal by similarity transformation, then there exists a matrix X Λ ( B ) with at least two different eigenalues. B 0 Proof. Let = 0 B. Let us comose the E 0 matrixxɶ = 0 E, where E and E are the identity matrices. Matrix X = SXS ɶ commutes with B and has two different eigenalues λ =, λ =. So, for the simultaneous reduction of two matrices to the bloc-diagonal form it is necessary and sufficient that the centralizer of the matrix to contain the matrix X with different eigenalues. To find the centralizer (or more recisely its basis), you can declare all the elements of the matrix X as unnown numbers and create a system of linear homogeneous algebraic equations corresonding to the matrix equations B Х = ХB, B Х = ХB. () We hae n equations with n unnowns. If n is small, then general solution of this system of equations can be made by nown methods. Comuting method of coing with large n is gien in [9]. Let W, W,, W r matrices form a basis of centralizer Λ(B ). If the ran r of the centralizer is equal to, then the whole centralizer consists of only matrices that are multile of the identity matrix. In this case, a reduction of matrices B to the bloc-diagonal form is imossible. If r >, then among the matrices W we choose a matrix X that has at least two different eigenalues. We draw u a matrix of similarity transformation of the column-ectors of the canonical basis of the matrix X. The secial case, where r >, but all matrices of the basis do not hae different eigenalues, is discussed below (see Section 6).

3 Physics Journal Vol., No., 05, The Method of Inariant Subsace The idea of the method is roosed in [] (Chater 7). Let us consider matrices B ( =, ) that are not reduced simultaneously to the bloc-diagonal form. Otherwise we would hae already done it by the method of the commutatie matrix. We want to find out, whether they could be reduced to the bloc-triangular form. 4.. Construction of Algebra The first ste of the method is the construction of algebra with unit ϕ ( B) generated by the initial matrices. You can do as follows []: firstly, we choose linearly indeendent elements of the matrix {E, B, B } and call them "assumed basis". Then we consider all the ossible roducts of these matrices. If the next roduct does not belong to the linear san of the "assumed basis", we add it to this set and consider the elements roducts of a new "assumed basis". We continue this rocess until none of the roducts goes beyond the linear san. The indication that the element does not belong to the linear san of the assumed basis" is that the addition of a new element gies a linearly indeendent set of elements. Verification of linear indeendence is ossible using rogram SLAU5 []. The ossibility of matrices reduction to the bloc-triangular form is equialent to the reducibility of algebra. The criterion of reducibility of algebra: the ran of the algebra ϕ(в ) is smaller than n, where n is matrices order. This follows from the Burnside theorem [0] (see also [6], Theorem '). 4.. Calculation of an Algebra Radical Ideal Theorem 3. If ran r of algebra ϕ(в ) is smaller than n and if centralizer Λ(B ) does not contain any matrix X with different eigenalues, then the algebra ϕ(в ) is nonsemisimle. Proof. The condition r < n means that the algebra ϕ(в ) is reducible [0, ]. A reducible algebra may be semisimle or non-semisimle. Algebra is not semisimle because the matrices (including {B }) are not reducible to the blocdiagonal form (if only they were reducible, we could do this by the method of commutatie matrix). Therefore the algebra is non-semisimle. A non-semisimle algebra has a nontriial radical ideal. There are formulas to find it []: the coordinates α = [α, α,, α r ] т of any radical ideal element in the basic set of the algebra satisfy the equation Dα = 0, D = {d i }, (3) where d i = S(W i W ), S(.) is the trace of matrix. {W i } is the basic set of the algebra. The general solution of equation (3) can be obtained by nown methods. You can, for examle, use the rogram SLAU5 []. Consequently, it is ossible to obtain a basis of radical ideal Finding of Inariant Subsace Let Z-set be intersection of all ernels of radical ideal elements of the algebra ϕ(в ). We can find Z-set (its basis) as a general solution of the corresonding system of algebraic equations. Theorem 4. Z-set is a subsace of sace U C. Proof. This set can be found with only elements of the basis. The calculation corresonds to finding a solution of the system of linear homogeneous algebraic equations. The general solution of this system, as we now, is a subsace. Theorem 5. If algebra is non-semisimle, then Z-set is a nontriial subsace. Proof. In this case, a non-zero radical ideal is a set of matrices Gτ ( ), where τ is the arameters ector. The radical ideal is a nilotent subalgebra because all its elements are nilotent (see [, 7, Theorem ]). Hence, : G ( τ ) 0, G + ( τ ) = 0. Let G be nonzero matrix of the set G ( τ ), and ξ its nonzero column. Then G ( τξ ) = 0 τ, since G( τ)( G ( τ )) = 0. So, the equation G ( τξ ) = 0 has nontriial solutions. Theorem 6. The subsace Z-set is an inariant with resect to the matrices {B }. Proof. Radical ideal Gτ ( ) of algebra ϕ(в ) is its ideal. Matrices B are members of algebra ϕ(в ). Hence, we obtain G( τ) Bξ = G( τ) ξ = 0 τ, where { : ( ), } ξ Z = ξ Gτξ = 0 τ, i.e. Z-set is inariant with resect to the set of matrices {B i }. Thus, for the case when the matrices are not reduced to the bloc-diagonal form, but the ran of the algebra ϕ(в ) is less than n, we hae a method of construction of nontriial subsace that is inariant with resect to these matrices Construction of the Transformation Matrix Theorem 7. The simultaneous reduction of a air of matrices to the bloc-triangular form is ossible if and only if there exists a nontriial inariant with resect to both matrices subsace U C.

4 57 Yuri N. Bazileich: The Simultaneous Reduction of Matrices to the Bloc-Triangular Form Proof. ( ). Let B B =, =,, 0 B 3 where B are matrices of order m. Then the set of ectors x y = 0, where m x C, is inariant with resect to matrices B ɶ m, i.e. if Vɶ x = y =, wherex C then 0 y Vɶ y Vɶ. Let V consist of all ectors of the form z = Sy, y Vɶ. From the inariance of the matrices, it follows thaty = y Vɶ Therefore,. z V Bz = SBS ɶ Sy = SB ɶ y = Sy V. ( ) Suose there exists a nontriial subsace V that is inariant with resect to the matrices be a basis of the subsace V. Let { s } to the basis of m+ B. Let { s s s },,..., m,..., s n be its addition U C. Let the columns of matrix S be the ectors { s,..., s n}. From the rules of matrix multilication, it follows that the equality AS = SA ɶ is an equialent to similarity transformation Aɶ = S AS, and this equality can be written as As n = aɶ s, =, n, (4) = where aɶ are the elements of the matrix A ɶ, A is an arbitrary matrix. The condition of inariance of the subsace V means that any element of the basis { s,..., s m} after multilication by any of the matrix B remains in the subsace V and, therefore, is a linear combination of elements{ s,..., s m} : it. We locate ectors as columns (see the roof of Theorem 7). Direct sum to the subsace can be found as a general solution x of the linear homogeneous algebraic equations s x = 0, =, m Τ. Here, T is a sign of transosition. To find the general solution, you can use the rogram SLAU5 []. 5. The Uniqueness Theorem There is the uniqueness theorem. Theorem 8. Let the matrices B, i i =,, be reduced to the bloc-triangular form Bi Bi... Bi l 0 B i... Bil i i =, i =, Bill by some similarity transformation and further reduction for each air of blocs {B, B } is imossible. If there exists another similarity transformation such that for the resulting blocs further reduction is imossible: B i B i... B il 0 B i... B il i = S BS i =, i =, B ill then l = l and we can determine the corresondence between the numbers of blocs such that the blocs B i are similar to blocs B : B = S B S. i( ) ( ) i i( ) ( ) Proof. This theorem follows from Theorem of Jordan Holder (see, also [], Theorem ). Bs m = βɶ s. = Comaring the last equality with (4) and taing into account the linear indeendence of ectorss, we obtain β ɶ = 0, = m +, n. These equations are erformed at all =, m. Consequently, the corresonding elements of matrices B ɶ are equal to zero, i.e. modified matrices hae the bloc-triangular form. We form the transformation matrix S from the basis ectors of this subsace and a subsace being a direct comlement to 6. Secial Case Let us consider the case when the basis of the centralizer contains more than one matrix (r > ), but each of these matrices has no different eigenalues. There is an assumtion that in this case all the matrices of the centralizer hae no different eigenalues and, accordingly, the initial matrices are not reduced to the bloc-diagonal form simultaneously. It turns out that this assumtion is alid if the algebra Λ( B ) has a ran r 3 and is not alid if ran r = 4 (see [], Theorem 6.6).

5 Physics Journal Vol., No., 05, This Section shows how in this case to reduce the matrices to the bloc-triangular form, without discussing the ossibility of reducing them to the bloc-diagonal form. Theorem 9. If the ran of the centralizer Λ( B ) of the matrices B is greater than one: r >, then the matrices B are reduced to the bloc-triangular form. Proof. Let W,, W r be the basis of algebra Λ( B ) and W =E. If W matrix hae two (or more) different eigenalues, then we can reduce initial matrices to the bloc-diagonal form by method of commutatie matrix. Otherwise eigenalue λ is unique. Therefore matrix G = W λ E is nilotent. Matrix G is nilotent and nonzero, therefore subsace L = { : G = } ξ ξ 0 is nontriial. Besides GBξ = BGξ = B0 = 0 ξ L, i.e. subsace L is inariant with resect of matrices B. Consequently, we can reduce matrices B to the bloc-triangular form in this case too (Theorem 7). We need to create a matrix G = W λ E and find the ectors { } s, s,..., s m as the basis of ernel matrix G to construct the transformation matrix S. Further construction is the same as in Subsection 4.4. Note. The condition r > is not necessary to reduce the matrices to the bloc-triangular form. 7. Examles 7.. The First Examle B = 0 0 0, B = (see[], examle.6) We use the method of commutatie matrix. All the elements x i of matrix X are considered as unnown. We constitute a system of algebraic equations corresonding to the matrix equations B Х = ХB, B Х = ХB. Its general solution is as follows: x = x ; x = x ; x = x x ; x = x = x = x where x and x are free unnowns. Or in other way X = x E + x = If x = 0, x = the eigenalues of X are: λ = λ = 0.5; λ 3 = 0.5. Then we obtain eigenectors of matrix X and build the transformation matrix S = The initial matrix is reduced to the bloc-diagonal form: = 0 0 0, = (5) Next, we consider the blocs B =, B. 0 0 = 4 5 For them a set of commuting matrices Λ( B ) is αe, where α is an arbitrary number. So reduction of these blocs to the diagonal form is imossible (Theorem ). We erify the ossibility of reducing by the method of inariant subsace. Matrices E, B and B are linearly indeendent. Products of any of these matrices to the identity matrix E belong to the linear san of the first three matrices. Matrix B is a member of this set too. We consider the roduct U = B B. Using the equality αe + β B + γ B + λ U = 0 we obtain: α = β = γ = λ = 0, i.e. matrix U does not belong to the linear san of the first three matrices. We obtain that r = 4 and the condition r < n is not erformed. Further simlification of the matrices is imossible (see Subsection 4.). Therefore the final result is the matrices (5). 7.. The Second Examle 0 B =, B 7 4 =. 7 4 (see[], examle 7.3). Direct erification shows that the corresonding centralizer consists only of matrices αe. Therefore, reduction of matrices B to diagonal form is imossible. Let us build algebra ϕ ( B ). Matrices E, B, B are linearly indeendent. Let us denote these matrices W, W, W 3 resectiely. Let us consider all ossible roducts of matrices W W and erify whether the resulting matrices are the linear combination of the original. Since multilication by W = E does not change the matrices, we consider roducts W W for 0, =, 3. We comute: W = 3 4. We erify whether the matrix is the linear combination of the first two ones: W = αw + βw. This equation corresonds to the system of equations. Its solution is: β = 3, α =. Consequently, the

6 59 Yuri N. Bazileich: The Simultaneous Reduction of Matrices to the Bloc-Triangular Form matrix is a linear combination of the matrices W and W. Next, we calculate: 7 4 WW 3 = = W3, WW 3 = = 6E + 3W + W3, 8 W3 = = 3 W3. We hae found that all roducts belong to the linear san of the matrices W, W, W 3. Consequently, these matrices form the basis of algebra ϕ ( B ). The number of elements of the basis r = 3, i.e. r < n. This means that the reduction to triangular form is ossible. Let us comose the matrix D = {S(W W )}. All roducts W W are already calculated. We obtain 3 3 D = Let us form the system of equations Dα = 0. The general solution of this system includes: α = 6α 3, α = 3α 3, where α 3 is a free ariable. Let α 3 =. We obtain α = [ 6 3 ] T. We comute the matrix G: G = = equations G ξ= 0 are of the form: The 4ξ + 4ξ = 0, Hence: 4ξ 4ξ = 0. ξ = ξ. We ut: ξ =. Therefore, the basis of Z-set consists of one ector: [ ] T ector [ ] T s = ξ =. This ector and the e = 0 are linearly indeendent. Therefore: S = 0. Then we obtain 0 S = ; ɶ ; B = S ES = E 0 0 B ɶ = = 0 0 ; =. = The Third Examle B = E, B = These matrices describe the motion of the system from [3]. It is clear, that we can reduce the matrix B to its Jordan form. But let us consider the aroach set forth aboe. Let us find matrix X that commutes with the initial matrices. As a result of calculations we obtain: X = αε, where α is an arbitrary arameter. Since matrix X has no different eigenalues, reduction of the initial matrices to the blocdiagonal form is imossible. Next, we use the method of inariant subsace. We act as in the reious examle. We obtain: r =. The condition r < n is erformed. Next: D =. The system of equations Dy = 0 taes the form: y + y = 0, y + y = 0. As a result, we obtain: y = radical ideal is a matrix G:. Basis of 0 G = = 3. Equation Gξ = 0 taes the form: ξ ξ = 0, ξ ξ = 0. The basis of general solutions of the system consists of the ector = s. This ector and the ector indeendent. Therefore, triangular matrices: ɶ, B = S ES = E S = s = are linearly. We obtain the = 0. Peculiarity of this examle is that here exists a non-comact grou of matrices commuting with the initial matrices.

7 Physics Journal Vol., No., 05, Generalizations The roblem of getting the best bloc-triangle form can be formulated in another way: to find a transformation matrix S such that the maximum of the orders of the diagonal blocs will be the lowest ossible. The uniqueness theorem imlies that by soling the roblem of getting the maximum number of blocs we simultaneously obtain the bloc haing the lowest ossible order. It is clear that there can be more than two initial matrices. In this case course of solution will hae no changes. Only the number of matrix equations () or the amount of the initial matrices for comosition of the algebra ϕ(в ) will be increased. Let us consider the roblem of reduction of matrices B, =, d to the bloc-triangular form by transformation B = HBS, where H and S are non-singular square matrices. ˆ This transformation is more effectie than the similarity transformation (). The roblem is soled in the case, where one of the initial matrices is nonsingular. Next, we need Theorem 0 by A.K. Loatin. Refined formulation and roof of the result is as follows (see [], Theorem 7.4). Theorem 0. There exists a similarity transformation reducing matrix B ( =, d) to the bloc-triangular form with l blocs on the main diagonal if and only if there exists a set of matrices G ( τ), ( τ C ) such that { } 0 ( ) G BL τ = 0 τ, n 0 C, L L L... L l where { : L = G ( ) = } Proof. ( ) Let matrices ξ t ξ 0 t C. B diml > + diml (6) ɶ be bloc-triangular. Let blocs B on the main diagonal be of the order m and d m = n. We choose a set of matrices ( ) = G( ) = SG ɶ ( ) S τ τ, where the matrices ( ) Gτ in the form G ɶ τ hae the same ind of bloc as B ɶ ; all the blocs of the ariable matrices G ɶ ( τ ) standing aboe the main diagonal are filled with arameters τ, τ,..., τ, and other elements of this matrix are zero. Then Т Т Т {,...,,... } L = ɶξ = x x 0 0, where x C m. It is clear that Gɶ ( τ) ξɶ = 0 ξ L. After the transformations equality is still alid. G( ) = SG ɶ ( ) S - τ τ, B SBS = ɶ, S ξ = ξ ɶ this ( ) The condition G ( τ) BL = G ( τ)( BL ) = 0 means that BL L matrix B., i.e. a subsace L is inariant with resect to the Let us denote l ( B, S) as the number of blocs on the main diagonal of the matrices B ɶ reduced to the bloctriangular form. Let lb ( ) = max l ( B, S). S: dets 0 Theorem. Let B, =, d be matrices. If B = E, then l( NB ) l( B ), where N is any non-singular matrix. Proof. Let N and S be the transformation matrices, whereby the initial matrices B are reduced to the bloctriangular form by formula ˆ B = S NBS and hae the maximum ossible number of blocs on the main diagonal. In this case matrices Dɶ ( ˆ = B) Bˆ hae the same blocdiagonal form. According to Theorem 0, there must be the matrices G ( τ), ( τ C ), such that { } ɶ ɶ ɶ 0 ( ) 0 ɶ τ ɶ ɶ = τ, G DL n 0 L L L L l C, diml > + where L { : = ɶ Gɶ ( ) ɶ = } ɶ ξ τ ξ 0. Let us erform a similarity transformation G( ) = SG ɶ ( ) S ɶ ɶ diml ɶ, D = SDS ɶ, ξ = Sξ. ɶ - τ τ, and relacement of ectors ξ ɶ into Equations (6) are retained. Moreoer, D = B: D = SDS ɶ = S ( S NS ) ( S NBS ) S = = SS N SS NBSS = B.? Using Theorem 0, we obtain l( B) l. For finding a transformation Bˆ = HBS with the greatest ossible number of blocs on the main diagonal it is sufficient to sole this roblem by a similarity transformation for suortie matrices C = B B, =, µ, µ = d. +

8 6 Yuri N. Bazileich: The Simultaneous Reduction of Matrices to the Bloc-Triangular Form 9. Conclusion Thus, the roblem is comletely soled. A method to bring a set of matrices to the best bloc-triangular form has been deeloed. This result is of ractical significance. This method can simlify a system of linear differential equations containing seeral matrices of coefficients [, 4]. Equations decouling to indeendent subsystems corresonds to reducing matrices to the bloc-diagonal form. Reduction of matrices to the bloc-triangular form corresonds to hierarchic (ertical) decouling. Thus, first subsystem does not contain ariables of other subsystems. Only ariables of the first and the second subsystems are resent in the second subsystem, etc. The number of such subsystems can be greater than at ordinary decouling. 0. Next Directions of Inestigation It would be helful to sole the following roblems as well. The solution of the same roblem for matrices oer other fields (real numbers, rational numbers, etc.). More detailed study of the secial case is considered in Section 6. Using the transformation Bˆ = HBS without requiring non-singularity of one of the matrices. Reduction to the best bloc-triangular form n n-matrix A and n n-matrix m-matrix B by transformation ˆ Aɶ S = AS, B. Here S and S are nonsingular square matrices of corresonding orders. This roblem and others, similar to it, are necessary for the hierarchic decouling of systems of equations with rectangular coefficient matrices (see [] Сhater 8 and []). References [] Bazileich Yu. N., Numerical decouling methods in the linear roblems of mechanics, Kyi, Nauoa Duma, 987 (in Russian). [] Bazileich Yu. N., Koroteno M.L. and Shets I.V., Soling the roblem on hierarchical decouling the linear mathematical models of mechanical systems, Tehnichesaya Mehania, 003, N, 35 4 (in Russian). [3] Bazileich Yu. N., The exact decouling of linear systems, Electronic Journal Issledoano Rossii, 006, 08, 8 90, htt:// (in Russian). [4] Drozd Yu. A., Tame and wild matrix roblems, Lect. Notes Math., 83, 4-58 (980). [5] Fermi Enrico, Notes on quantum mechanics/ The Uniersity of Chicago Pres [6] Loatin A.K., The algebraic reducibility of systems of linear differential equations. I, Diff. Uran., 968, V.4, (in Russian). [7] Yauboich E.D., Construction of relacement systems for a class of multidimensional linear automatic control systems, Iz. Vuzo, Radiofizia, 969, V., N 3, (in Russian). [8] Gantmacher F. R., The Theory of Matrices / Chelsea: New Yor [9] Bazileich Yu. N., Buldoich A.L., Algorithm for finding the general solution of the system of linear homogeneous algebraic equations in the case of ery large-scale sarse matrix coefficients // Mathematical models and modern technology. Coll. scientific. tr. / NAS. Institute of Mathematics. - Kyi, 998., 3 (in Russian). [0] Van Der Waerden Algebra. ols. I and II. Sringer-Verlag. [] Chebotario N.G., Introduction to theory of algebras, Moscow: Editorial URSS, 003 (in Russian). [] Belozero V.E., Mozhae G.V., The uniqueness of the solution of roblems of decouling and aggregation of linear systems of automatic control // Teoriya slozhnyih sistem i metodyi ih modeliroaniya. M: VNIISI, (in Russian). [3] Bazileich Yu. N. Hidden Symmetry Exosure. The Mechanical Systems with the Hard Structure of Forces // Proceedings of Institute of Mathematics of NAS of Uraine. V. 50. Part 3. Kyi: Institute of Mathematics of NAS of Uraine, 004 / P [4] Palosy Yu. N., Smirnoa T.G., The roblem of decomosition in the mathematical modeling. M.: FAZIS, 998. VI + 66 (in Russian).