Factorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution

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1 Goba Journa of Pure and Appied Matheatics ISSN Voue 13 Nuber 10 (017) pp Research India Pubications Factorizations of Invertibe Syetric Matrices over Poynoia Rings with Invoution MI Kucha Departent of Matheatics Lviv Poytechnic Nationa University Lviv Ukraine Abstract By eans of the notions of infinite eeentary divisors dua and generaized dua atrix poynoias we find necessary and sufficient conditions for the existence of factorizations of invertibe syetric atrices over rings of poynoias with invoution Keywords: poynoia atrices infinite eeentary divisors dua and generaized dua poynoia atrix adissibe and inadissibe factorizations AMS Subject Cassification: 15A3 I INTRODUCTION One of the ost iportant probes in the atrix theory is the probe of factorization of syetric atrices over a poynoia ring with invoution cassification and investigation of the structure of there divisors [ ] It is known that agebraic probes with productions of this type occur in the theory of differentia equations the noninear contro theory the optia contro and integrated systes of cassica echanics and so have soe practica vaue [1 8] In particuar [7 9] deat the probe of factorization of invertibe syetric atrices over a poynoia rings C[ x ] Such factorization of atrix carried by aocation of diagona and eiinate of zero on the diagona In this paper we present a sipe ethod of finding the factorization of invertibe syetric atrices over a poynoia ring which uses the notions of dua poynoia atrix [3] and generaized dua poynoia atrix [6] and their factorizations Our resuts is based on the theory presented in recent papers [ ] In particuar in [5 6 10] using the notions of infinite eeentary divisors dua poynoia atrix and generaized dua poynoia atrix were found conditions of existence factorization of a singuar atrix poynoias In

2 7074 MI Kucha the [11] was investigated the probe of finding factorization syetric reguar atrices over poynoia rings with invoution II PRELIMINARIES Let K C[ x] be the poynoia ring with invoutions defined in [7]: p p ( ) i i a i x ai x i i p p ( ) i i a i x a i ( i i p p ( ) i a i x a i ( i i i and extended to the atrix ring M n (K) as foows: A atrix A ( is caed syetric if a ij ( a ji ( A ( A decoposition of invertibe syetric atrix A ( is caed factorization where B ( is invertibe atrix A ( B( CB( (1) (det C 0) Let us consider nonsinguar poynoia atrix C C is a nonsinguar diagona atrix A i ( x ) A i x i0 where A i M n (C) The poynoia atrix A ( is caed singuar (reguar) if det A 0 0 ( det A 0 0 ) [] If A 0 E ( E atrix identity) then the poynoia atrix A ( is caed unita [4]

3 Factorizations of Invertibe Syetric Matrices over Poynoia Rings with Invoution 7075 Let us define the dua poynoia atrix A ( x ) of A ( as in [3]: A i ( x ) A i x i0 The infinite eeentary divisors of A ( are defined as the finite eeentary divisors of A ( x ) at x 0 ie as the finite eeentary divisors of A ( that have the for: x j j 0 It is easy to see that if the poynoia atrix A ( is invertibe then dua poynoia x det atrix A n ( ) is reguar atrix with characteristic poynoia x Let us denote ( where ( Q( S A the Sith for of the atrix A ( x ) S A ( P( Q( () P are invertibe atrices over K A factorization of a syetric atrix A ( x ) is caed adissibe if the Sith for of the atrix A ( x ) is the product of Sith fors of the factors in the decoposition (1) Otherwise a factorization of a syetric atrix A ( x ) is caed inadissibe In the [5] studied the probe of aocating specia divisor fro singuar atrix poynoia It has been proved that a necessary and sufficient condition for the decoposition B( C( where B ( is singuar poynoia atrix of degree r with the Sith for ( diag ( 1( ( ) and syste of n infinite eeentary divisors x x x n n n n i1 i1 oreover deg deg B( deg C( is the condition 0 1 i rn deg i ( det M ( ) 0 (3) V ( ) P( E Ex Ex r1

4 7076 MI Kucha where ( ) ( 1 ( x diag x ) 1 x x n n( x )) P ( is arbitrary invertibe atrix of the reation () the atrix V ( ) is defined in [4] Condition (3) is necessary condition for the factorization (1) but not sufficient In the present parer we found necessary and sufficient conditions for the existence of the factorization (1) of a syetric invertibe atrix A ( over K where B GL (K) ( n deg B( III MAIN RESULTS Theore 1 For a syetric invertibe atrix ( A over K there exists a factorization (1) where B ( is invertibe atrix with the syste of infinite eeentary divisors n x 1 x x n 0 1 n n i and i1 a nonsinguar diagona atrix if and ony if the syetric atrix V ( ) P( P( V ( ) is siutaneousy divisibe by ( fro the eft and by C C is ( fro the right for for certain vaues of the paraeters of the atrix V ( ) for which hods the condition (3) where ( diag ( x 1 x n ) P ( is arbitrary invertibe atrix of the reation () Proof Necessity foows fro Theore 1 [10] Sufficiency Ву Theore 1 [11] there is a factorization of the dua poynoia reguar atrix A ( x ) of ( where 1( x ) A : ( B ( GB ( A 1 1 (4) and B is an unita atrix of degree with the Sith for ( G G is a nonsinguar atrix The atrix G is *congruent atrix for the invoution ( ) (congruent atrix for invoutions ( ) ) ( ) to the atrix inertia

5 Factorizations of Invertibe Syetric Matrices over Poynoia Rings with Invoution 7077 I ( k G ie G TI G) T G ) diag { E k E n } where k is a nuber of positive eigenvaues of atrix where ( where GL n (C) T Then fro the reation (4) we obtain a factorization A ( B( C B( (5) B ( B ( T is a reguar atrix of degree ( B ( x ) and C I ( ) 1 1 G Consider the dua poynoias of ( ) 1 A x ( is the Sith for of x B and B ( ) fro the reation (5) Thus we get the factorization (1) of invertibe atrix A ( over K where B ( is invertibe atrix over K with a syste of infinite eeentary divisors 1 x x n and C C1 The theore is proved x As a cororay we have Cororay For a syetric invertibe atrix factorization (1) where A GL (K) there exists a ( n 1 B ( GLn (K) deg B( deg if and ony if A ( x of A ( there exists a factorization for the dua syetric atrix ) There is an iportant probe of the factorization (1) of the invertibe syetric atrix A ( over K of degree where GL (K) condition deg B ( n deg B( The A ( deg B( deg B( is not hods in the factorization (1) Therefore we introduce into consideration the notion of generaized dua poynoia i Let f x a i x ( ) be a poynoia in particuar a poynoia atrix i0 The generaized dua poynoia f ( x ) of f ( for degree r ( r ) is define as [6]: f ( x ) i0 a i x ir where r N

6 7078 MI Kucha If deg f ( it is obvious that f ( x f r N x A (x is invertibe over K then A ( x is a reguar atrix poynoia If A ( is a A ( x is a It is easy to see that if a poynoia atrix ) generaized dua poynoia ) r 1 syetric atrix poynoia then its generaized dua poynoia ) syetric for an even nuber r Further assue that r is an even nuber Given that r Ex A ( r N we can easiy show the foowing resut Proposition Let the Sith for ( for () Then the Sith for ( A ( for degree r is where P ( GL (K) S A of dua poynoia A ( x ) of ( S of the generaized dua poynoia ( x ) A S ( P( Q( ) A x Q ( n are atrices fro the reation () A has the A of Theore For a syetric atrix A ( GLn (K) there exists a factorization C (1) where B ( GLn (K) C is a nonsinguar diagona atrix if and ony if there exists the inadissibe factorization of generaized dua poynoia A ( x ) of A ( for degree r where r B x B deg ( ) deg ( the atrix ( x ) the dua poynoia of B ( Proof Necessity Let for a atrix poynoia ( n factorization (1) and r deg B( deg B( deg Let us consider the generaized dua poynoia A ( x ) of ( B is A GL (K) there exists a A for degree r Then fro the reation (1) we obtain the factorization of a reguar atrix poynoia A ( x ) We ust to prove that such factorization of atrix A ( x ) is inadissibe

7 Factorizations of Invertibe Syetric Matrices over Poynoia Rings with Invoution 7079 Suppose by contradiction that there exists adissibe factorization for A ( x ) : A ( B ( C1B ( and where 1 x ) n( r) S ( 1( I1( deg ( ) A 1 x B ( with the Sith for ( ) Since ( ) r S x Ex S ( ) 1 x r ast reations we can aocate the factors Ex and A x then fro A r Ex respectivey: r r B1 ( Ex B( ( ) 1 x Ex ( It foows that there exists adissibe factorization of atrix A ( x ) where B ( is the atrix of degree with the Sith for ( Thus we obtain a contradiction since deg B ( deg B( Sufficiency Let there exists inadissibe factorization of the generaized dua poynoia A ( x ) of A ( for degree r ie A ( B ( C1B ( (6) where deg A ( deg B ( deg B ( Let us consider the dua poynoia A ( of A ( x ) Then fro the reation (6) we obtain the factorization of invertibe atrix poynoia A ( over K where deg Last inequaity hods because the coefficients of zero The theore is proved A ( deg B( deg B( x r1 x 0 in the atrix ( x ) A are REFERENCES [1] Andreev V A Shepeyaviy A M Synthesis of optia contros for puseapitude systes in the probe of iniization of the ean vaue of a functiona of quadratic type // Sib Mat Zh No pp (in Russian)

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