m k,k orthogonal on the unit circle are connected via (1.1) with a certain (almost 1 ) unitary Hessenberg matrix

Transcription

1 A QUASISEPARABLE APPROACH TO FIVE DIAGONAL CMV AND COMPANION MATRICES T. BELLA V. OLSHEVSKY AND P. ZHLOBICH Abstract. Recent work in the characterization of structured matrices in terms of the systems of polynomials (and specifically the recurrence relations they satisfy) related as characteristic polynomials of principal submatrices is furthered in this paper. It has been shown previously that quasiseparable structure (essentially low rank blocks up to but excluding the main diagonal) is particularly useful in providing such recurrence relations characterizations. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices unitary Hessenberg matrices (related to Szegö polynomials) and semiseparable matrices as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify known results. Previous work has focused on characterizing (H ) quasiseparable matrices matrices with order one quasiseparable structure that are also upper Hessenberg. This restriction permits a useful bijection between the sets of matrices and systems of polynomials and thus the results derived are complete characterizations. In this paper the authors introduce the concept of a twist transformation and use them to explain the relationship between (H ) quasiseparable matrices and the subclass of ( ) quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results explain the discoveries of Cantero Fiedler Kimura Moral and Velázquez of five diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. Key words. quasiseparable matrices semiseparable matrices CMV matrices Kimura unitary Hessenberg matrices Fiedler matrices banded matrices five diagonal matrices companion matrices well free matrices orthogonal polynomials Szegö polynomials twist transformation. AMS subject classifications.. Introduction. Various polynomial systems {r k (x)} n k=0 are often associated with Hessenberg matrices H = [m ij ] n ij= as scaled characteristic polynomials of principal submatrices of H; that is via the relation r 0 (x) = λ 0 r k (x) = λ 0 λ... λ k det(xi H k k ) k =... n. (.) Moreover the relation (.) establishes a bijection [5] if λ k = parameters so m kk and λ 0 λ n are two {r k (x)} n k=0 {H λ 0 λ n }. (.2).. From Hessenberg to five diagonal matrices. Two examples. It is widely known that Szegö polynomials {φ # k (x)}n k=0 orthogonal on the unit circle are connected via (.) with a certain (almost ) unitary Hessenberg matrix ρ 0ρ ρ 0µ ρ 2 ρ 0µ µ 2ρ 3 ρ 0µ µ 2µ 3 µ n ρ n µ ρ ρ 2 ρ µ 2 ρ 3 ρ µ 2 µ 3 µ n ρ n M = 0 µ 2 ρ 2ρ 3 ρ 2µ 3 µ n ρ n (.3) 0 0 µ n ρ n ρ n Department of Mathematics University of Rhode Island Kingston RI 0288 USA. (tombella@math.uri.edu). Department of Mathematics University of Connecticut Storrs CT USA. (olshevsky@math.uconn.edu) (zhlobich@math.uconn.edu). Throughout the paper matrices referred to as unitary Hessenberg are almost unitary differing from unitary in the last column. Specifically M = UD for a unitary matrix U and diagonal matrix D = diag{... ρ n }.

2 2 T. Bella V. Olshevsky and P. Zhlobich where ρ k are reflection coefficients 2 and µ k are complementary parameters. The details on this relation can be found in [ ]. The matrix M has a rather dense structure in comparison with the tridiagonal Jacobi matrix [ 5 25] for orthogonal polynomials on the real line. However the bijection (.2) implies that for a given system of Szegö polynomials the only Hessenberg matrix related to that system via (.) is M. The situation is much different if we do not restrict the matrix to the class of strictly upper Hessenberg matrices. It was found first by Kimura [28] and independently by Cantero Moral and Velázquez [6 7 8] that Szego polynomials are also related via (.) (with λ k = µ k ) to the following five diagonal matrix: ρ 0ρ ρ 0µ 0 µ ρ 2 ρ ρ 2 µ 2 ρ 3 µ 2 µ 3 µ µ 2 ρ µ 2 ρ 2ρ 3 ρ 2µ 3 0 K = 0 µ 3ρ 4 ρ 3ρ 4 µ 4ρ 5 µ 4µ 5 (.4) µ 3 µ 4 ρ 3µ 4 ρ 4ρ 5 ρ 4µ 5 0 which has been called a CMV matrix since the paper [6] triggered deep interest in the orthogonal polynomials community. It is reputed that CMV matrices are better than unitary Hessenberg matrices in studying of properties of polynomials orthogonal on the unit circle (mostly because of its banded structure). Shortly after the discovery of CMV matrices other non Hessenberg matrices related to important systems of polynomials via (.) were discovered. Consider the well known companion matrix m m 2 m n m n C = (.5) The characteristic polynomials {p k (x)} n k=0 of its leading submatrices are the so called Horner polynomials. It was shown by Fiedler [40] that the five diagonal matrix m m m 3 0 m 4 F = (.6) 0 m 5 0 m 6 is also related to the same set of Horner polynomials..2. Quasiseparable approach. Twist transformation. What do CMV matrices (.4) and Fiedler matrices (.6) have in common? It turns out that both of them belong to the class of ( ) quasiseparable matrices defined next. Definition. (Rank definition of ( ) quasiseparable matrices). A matrix A is called ( ) quasiseparable (i.e. order one quasiseparable) if max rank A( : i i + : n) = max rank A(i + : n : i) =. i n i n 2 Reflection coefficients are also known in various contexts as Schur parameters [36] Verblunsky coefficients [37].

3 A quasiseparable approach to five diagonal CMV and companion matrices 3 Indeed every submatrix A( : i i + : n) or A(i + : n : i) of CMV and Fiedler matrices consists of at most two nonzero elements in the same row or column and therefore rank A( : i i + : n) = rank A(i + : n : i) = for all i =... n. It can be also shown that unitary Hessenberg matrices (.3) and companion matrices (.5) are also ( ) quasiseparable we refer to [9] for details. Eidelman and Gohberg in [9] gave an alternative definition of ( ) quasiseparable matrices in terms of small number of parameters they are described by. Such sparse representations are often at the heart of fast algorithms involving this and similar classes of structured matrices. Definition.2 (Generator definition of ()-qs matrices). A matrix A is called ( ) quasiseparable if it can be represent in the form d g h 2 g b 2h 3 g b 2... b n h n p 2 q d 2 g 2 h 3 g 2 b 3... b n h n p 3 a 2 q p 3 q 2 d 3 g 3 b 4... b n h n d n g n h n p n a n... a 2 q p n a n... a 3 q 2 p n a n... a 4 q 3 p n q n d n where the parameters {q k a k p k d k g k b k h k } all scalars are called generators of A. One of many useful properties of ( ) quasiseparable matrices is the existence of two term recurrence relations for polynomials related to them via (.). Theorem.3. [23] Let {r k (x)} n k=0 be a system of polynomials related to a ( ) quasiseparable matrix A via (.). Then they satisfy two term recurrence relations F0 (x) 0 = r 0 (x) λ 0 Fk (x) r k (x) [ ak b = λ k x c k q k g k k p k h k x d k ] Fk (x) (.7) r k (x) where c k = d k a k b k q k p k b k g k h k a k. Remark.4. The choice of generators of an ( ) quasiseparable matrix is not unique. What one can get immediately from this theorem is that the interchange of lower and upper generators as in a k b k p k h k q k g k (.8) for some k does not change the recurrence relations (.7) and hence does not change polynomials {r k (x)} n k=0. We propose to call an operation described by (.8) a twist transformation. We next show that both CMV and Fiedler matrices can be obtained via twist transformations from unitary Hessenberg and companion matrices respectively. Example.5 (Unitary Hessenberg and CMV matrices). By comparing ( ) - quasiseparable generators of unitary Hessenberg (Table.) and CMV (Table.2) matrices we conclude that the second is obtained from the first via twist transformations for even indices. Example.6 (Companion and Fiedler matrices). Similarly comparing Tables.3 and.4 one can see that the Fiedler matrix is obtained from the companion matrix via twist transformations for odd indices k >.

4 4 T. Bella V. Olshevsky and P. Zhlobich Table. Generators of unitary Hessenberg matrix d k a k b k q k g k p k h k ρ k ρ k 0 µ k µ k ρ k µ k ρ k Table.2 Generators of CMV matrix k d k a k b k q k g k p k h k odd ρ k ρ k 0 µ k µ k ρ k µ k ρ k even ρ k ρ k µ k 0 ρ k µ k µ k ρ k The invariance of systems of polynomials under twist transformation together with Examples.5 and.6 explains why unitary Hessenberg and CMV as well as companion and Fiedler matrices share the same systems of characteristic polynomials..3. Main results and structure of the paper. As we have mentioned already unitary Hessenberg and companion matrices are both strictly upper Hessenberg and ( ) quasiseparable. Such matrices have often been called (H ) quasiseparable (see Definition 2.5). In Section 2 of the present paper we study matrices obtained from (H ) quasiseparable matrices via twist transformation (which we call twisted (H ) quasiseparable matrices). We also show that every (H ) quasiseparable matrix can be transformed to a certain five diagonal matrix (one of twisted (H ) quasiseparable matrices) via twist transformations. The next part of the paper is devoted to the study of recurrence relations for (scaled) characteristic polynomials of principal submatrices of five diagonal twisted (H ) quasiseparable matrices. In the recent paper [9] authors derived specific recurrence relations for various subclasses of (H ) quasiseparable matrices. Moreover because of the bijection (.2) they have obtained a full characterization of subclasses of (H ) quasiseparable matrices via recurrence relations satisfied by polynomials related to them via (.). We give a brief survey of the results of [9] in Section 3 in order to exploit them in Section 4 in connection with five diagonal matrices. In particular we derive the class of five diagonal matrices which are connected to polynomials satisfying three term recurrence relations r 0 (x) = α 0 r (x) = (α x + β ) r 0 (x) r k (x) = (α k x + β k ) r k (x) + (γ k x + δ k ) r k 2 (x) α k 0 k = 2... n. (.9) It was shown by Geronimus [24] that under the additional restriction of ρ k 0 for every k the corresponding Szegö polynomials {φ # k (x)}n k=0 satisfy three term recurrence relations φ # 0 (x) = µ 0 φ # k (x) = [ µ k x + φ # (x) = (x φ # 0 µ (x) + ρ ρ 0 φ # 0 (x)) ] φ # k (x) ρ k ρ k µ k ρ k ρ k µ k µ k x φ # k 2 (x) (.0) which are of type (.9). One can also check that the Horner polynomials related to the Fiedler matrix (.6) satisfy the recurrence relations ( p k (x) = x + m ) k p k (x) m k x p k 2 (.) m k m k

5 A quasiseparable approach to five diagonal CMV and companion matrices 5 Table.3 Generators of companion matrix k d k a k b k q k g k p k h k k = m k m k Table.4 Generators of Feidler matrix k d k a k b k q k g k p k h k k = m k > odd m k even m k under the restriction m k 0 for every k. Recurrence relations (.) are of type (.9) as well. Hence both five diagonal CMV matrices (.4) and Fiedler matrices (.6) belong to the new class of five diagonal matrices related to polynomials satisfying three term recurrence relations. This result as well as other results on polynomials related to five diagonal matrices are presented in Section 4. Finally in Section 5 we derive a nested decomposition of twisted (H ) quasiseparable matrices which can also be used to obtain them from the original (H ) - quasiseparable matrices. 2. Twist transformations and twisted (H ) quasiseparable matrices. 2.. Twist transformations. A system of polynomials can be related to many distinct ( ) quasiseparable matrices (Definition (.2)) via (.). For instance a nonsymmetric ( ) quasiseparable matrix and its transpose share the same system of polynomials. In this section we show how for a given ( ) quasiseparable matrix one can obtain another ( ) quasiseparable matrices related to the same system of polynomials as the original one. Definition 2. (Twist transformation). We say that an n n ( ) quasiseparable matrix à having generators { p k q k ã k g k h k b k d k } is obtained via twist transformation from another n n ( ) quasiseparable matrix A with generators {p k q k a k g k h k b k d k } if there exists a set K { 2... n} such that q = g g = q d = d if K p n = h n hn = p n dn = d n if n K (2.) p k = h k q k = g k ã k = b k if k K hk = p k g k = q k bk = a k dk = d k and all other generators of à and A are equal. Additionally if K contains a single index we call the transformation an elementary twist transformation. In other words à is obtained from A via the interchange of lower and upper generators a k b k p k h k q k g k for some subset of indices k. This is why we propose to call the operations of (2.) twist transformations.

6 6 T. Bella V. Olshevsky and P. Zhlobich The significant feature of the twist transformation is that it transforms one ( ) quasiseparable matrix into another while preserving the coefficients of the recurrence relations (.7) and thus also preserving the characteristic polynomials of all of their submatrices. The next theorem exploits this fact. Theorem 2.2. The system of polynomials related to a ( ) quasiseparable matrix A is invariant under twist transformations. Proof. It suffices to prove the theorem for an elementary twist transformation with K = {k}. Let à be the matrix obtained from A via (2.) and {r k(x)} n k=0 and { r k (x)} n k=0 be the system of polynomials related to A and à respectively. Considering the recurrence relations (.7) for polynomials related to ( ) quasiseparable matrices and noticing that ã k bk = a k b k p k hk = p k h k dk = d k d k ã k bk q k p k bk g k hk ã k = d k a k b k q k p k b k g k h k a k. we conclude that both systems of polynomials {r k (x)} n k=0 and { r k(x)} n k=0 satisfy the same recurrence relations and hence coincide. Corollary 2.3. Examples.5 and.6 show that CMV matrices (.4) and Fiedler matrices (.6) are obtained via twist transformations from unitary Hessenberg matrices (.3) and companion matrices (.5) respectively. Hence unitary Hessenberg matrices and CMV matrices share the same systems of characteristic polynomials as do companion matrices and Fiedler matrices Corollary 2.4. For an arbitrary ( ) quasiseparable matrix A of size n there are 2 n (possibly not distinct) matrices obtained from A via twist transformations related to the same system of polynomials as A Twisted (H ) quasiseparable matrices. Following [9] we define the class of matrices which are both strictly 3 upper Hessenberg and ( ) quasiseparable. The definition like Definition.2 above is given in terms of generators see [9] for an equivalent definition in terms of ranks. Definition 2.5 (Generator definition of (H ) quasiseparable matrices). A matrix A is called (H ) quasiseparable if it can be represented in the form d g h 2 g b 2h 3 g b 2... b n h n q d 2 g 2 h 3 g 2 b 3... b n h n 0 q 2 d 3 g 3b 4... b n h n A = (2.2) qn 2 d n g n h n 0 0 q n d n where the parameters {q k 0 d k g k b k h k } are called generators of A. Remark 2.6. Comparing Definitions.2 and 2.5 one can easily see that an ( ) quasiseparable matrix is (H ) quasiseparable if and only if it has a choice of generators such that a k = 0 p k = q k 0. It is easy to check that both unitary Hessenberg matrices (.3) and companion matrices (.5) are (H ) quasiseparable (in fact the generators listed for them in Tables. and.3 demonstrate this fact). As we have seen CMV matrices (.4) and Fiedler matrices (.6) can be obtained from them via twist transformations. In order 3 i.e. having nonzero subdiagonal elements.

7 A quasiseparable approach to five diagonal CMV and companion matrices 7 to generalize these results we define next the entire class of matrices which can be obtained from (H ) quasiseparable matrices via twist transformations. Definition 2.7 (Twisted (H ) quasiseparable matrices). A ( ) quasiseparable matrix A is called twisted (H ) quasiseparable if it can be obtained from an (H ) quasiseparable matrix via twist transformations. Performing the twist transformation of the matrix (2.2) explicitly one can give the following alternative definition of twisted (H ) quasiseparable matrices in terms of their generators. Definition 2.8 (Generator definition of twisted (H ) quasiseparable matrices). A ( ) quasiseparable matrix A is twisted (H ) quasiseparable if and only if it has a choice of generators {p k q k a k g k h k b k d k } such that q 0 or g 0 a k = 0 q k 0 p k = or b k = 0 g k 0 h k = k = 2... n p n = or h n =. For an arbitrary (H ) quasiseparable matrix A with given generators according to Corollary 2.4 there are exist 2 n (possibly not distinct) twisted-(h ) quasiseparable matrices related to the same polynomial system as A. But it is feasible to distinguish them using the so-called pattern defined next. Definition 2.9 (Pattern of twisted (H ) quasiseparable matrices). For an arbitrary twisted (H ) quasiseparable matrix A we will say that a sequence of binary digits (i i 2... i n ) is the pattern of A if A can be transformed to some (H ) quasiseparable matrix H by applying the twist transformation for k K for each i k =. Equivalently (i i 2... i n ) is the pattern of A if there exist generators of A satisfying q 0 if i = 0 g 0 if i = a k = 0 q k 0 p k = if i k = 0 b k = 0 g k 0 h k = if i k = p n = if i n = 0 h n = if i n =. (2.3) Under these conditions we will also say that A = H(i i 2... i n ). Example 2.0. In accordance to this definition any (H ) quasiseparable matrix H of size n is H( ) and its transpose is H(... ). }{{}}{{} n n Example 2.. Comparing the generators of unitary Hessenberg matrices (Table.) and CMV matrices (Table.2) it is easy to see that CMV matrices have pattern ( ). A similar observation shows that Fiedler matrices have pattern ( ). Remark 2.2. Let H be an (H ) quasiseparable matrix specified by its generators {q k d k g k b k h k }. Then the matrices H( ) and H( )

8 8 T. Bella V. Olshevsky and P. Zhlobich are five diagonal. In particular d g 0 q h 2 d 2 q 2 h 3 q 2 b 3 q b 2 g 2 d 3 g 3 0 H( ) = 0 q 3 h 4 d 4 q 4 h 5 q 4 b 5 q 3b 4 g 4 d 5 g 5 0 and H( ) = H( ) T. Thus for every (H ) quasiseparable matrix there always exist five diagonal twisted (H ) quasiseparable matrices having the same system of characteristic polynomials. More details on five diagonal matrices will be given in Section A survey of [9] results for (H ) quasiseparable polynomials. In the present section we briefly describe the main results of [9] in order to use them extensively in Section A bijection between strictly Hessenberg matrices and polynomial systems. Let H n be the set of strictly upper Hessenberg n n matrices λ 0 and λ n } be two nonzero parameters and P n be the set of polynomial systems {r k } n k=0 with deg r k = k. We next demonstrate that there is a bijection between the triple (H n λ 0 λ n ) and P n. Indeed given a polynomial system {r k } n k=0 satisfying deg r k = k there exist unique n term recurrence relations of the form r 0 (x) = m 00 x r k (x) = m kk r k (x) m k k r k (x) m 0k r 0 (x) m kk 0 k =... n. (3.) This formula represents x r k in the space of all polynomials of degree at most k in terms of {r j } k j=0 which form a basis in that space and hence these coefficients are unique. Forming a matrix H H n and parameters λ 0 and λ n from these coefficients of the form m 0 m 02 m 03 m 0n m m 2 m 3 m n. H = 0 m 22 m λ 0 = λ n = (3.2) m.. 00 m nn mn 2n 0 0 m n n m n n it is clear that there is a bijection between (H n λ 0 λ n ) and P n as they share the same unique parameters. Furthermore it was shown in [32] that the strictly upper Hessenberg matrix H defined in (3.2) and the polynomial system (3.) are related via (.) with λ k = m kk. This shows the desired bijection (.2). The matrix H in (3.2) is usually called confederate for the system of polynomials (3.). To conclude for an arbitrary matrix H and scaling factors {λ k } n k=0 there exists a unique system of polynomials related to it via (.). But the converse is of course not true. However a bijection does exist if we restrict our attention to strictly upper Hessenberg matrices.

9 A quasiseparable approach to five diagonal CMV and companion matrices Well-free polynomials and three term recurrence relations. Consider the general three term recurrence relations r 0 (x) = α 0 r (x) = (α x + β ) r 0 (x) r k (x) = (α k x + β k ) r k (x) + (γ k x + δ k ) r k 2 (x) α k 0. (3.3) These recurrence relations can be treated as the generalized version of the recurrence relations r k (x) = (α k x + β k )r k (x) + γ k r k 2 (x) α k 0 γ k > 0 (3.4) satisfied by polynomials orthogonal on the real line. Theorem 3. (General three term recurrence relations). A polynomial system {r k (x)} n k=0 satisfies three term recurrence relations (3.3) if and only if there exists an (H ) quasiseparable matrix H with the set of generators {q k d k g k b k h k 0} related to it via (.) with λ k = q k. Moreover conversion formulas between generators and recurrence relations coefficients are given in Table 3.. Table 3. Conversion formulas: three term r.r. coefficients quasiseparable generators. quasiseparable generators q k d k g k b k h k α k α k β k +γ k α k α k γ k+d k +δ k α k+ γ k+ α k+ three term r.r. coefficients α k β k γ k δ k q k q k b k d k q k b k q k d k b k g k q k+ Remark 3.2. (H ) quasiseparable matrices with the restriction h k 0 on the generators were called well-free in [9]. Therefore polynomials satisfying (3.3) are also called well-free polynomials Semiseparable polynomials and Szegö type two term recurrence relations. Another interesting family of recurrence relations for polynomials considered in [9] are the so-called Szegö type two term recurrence relations of the form G0 (x) β0 = r 0 (x) δ 0 [ ] Gk (x) αk β = k G k (x) (3.5) r k (x) γ k δ k (x + θ k ) r k (x) with α k δ k β k γ k 0 δ k 0 and G k (x) being auxiliary polynomials. These recurrence relations generalize those satisfied by Szegö polynomials (polynomials orthogonal on the unit circle) of the form φ0 (x) φ # 0 (x) = ρ 0 µ 0 φk (x) φ # k (x) = ρ φk k (x) µ k ρ k xφ # k (x). (3.6) This justifies the name Szegö type. Theorem 3.3 (Szegö type two term recurrence relations). A polynomial system {r k (x)} n k=0 satisfies two term recurrence relations (3.5) if and only if there exists

10 0 T. Bella V. Olshevsky and P. Zhlobich Table 3.2 Conversion formulas: Szegö two term r.r. coefficients quasiseparable generators. quasiseparable generators q k d k g k b k h k δ k θ k + β k γ k δ k δ k β k (α k δ k β k γ k ) δ k δ k (α k δ k β k γ k ) γ k Szegö type r.r. coefficients α k β k γ k δ k θ k b k q k + g k+q k+ b k+ h k g k+ q k+ b k+ q k h k q k d k g k h k an (H ) quasiseparable matrix H with the set of generators {q k d k g k b k 0 h k } related to it via (.) with λ k = q k. Moreover conversion formulas between generators and recurrence relations coefficients are given in Table 3.2. Remark 3.4. (H ) quasiseparable matrices with the restriction b k 0 on the generators were called semiseparable in [9]. Therefore polynomials satisfying (3.5) are also called semiseparable polynomials Quasiseparable polynomials and EGO type two term recurrence relations. The authors of [9] established that the class of polynomials related to (H ) quasiseparable matrices (Definition 2.5) are characterized as those satisfying EGO type two term recurrence relations F0 (x) r 0 (x) = [ 0 θ0] Fk (x) r k (x) [ βk γ = k δ k θ k x + ε k ] [ Fk (x) r k (x) ] (3.7) with auxiliary polynomials F k (x). Theorem 3.5 (EGO type two term recurrence relations). A polynomial system {r k (x)} n k=0 satisfies two term recurrence relations (3.7) if and only if there exists an (H ) quasiseparable matrix H with the set of generators {q k d k g k b k h k } related to it via (.) with λ k = q k. Moreover conversion formulas between generators and recurrence relations coefficients are given in Table 3.3. Table 3.3 Conversion formulas: EGO-type r.r. coefficients quasiseparable generators. quasiseparable generators q k d k g k b k h k θ k ε k θk γ k β k δ k θk EGO type r.r. coefficients β k γ k δ k θ k ε k b k g k h k q k q k d k qk Remark 3.6. Due to the bijection established by Theorem 3.5 it was proposed in [9] to call polynomials satisfying the recurrence relations (3.7) (H ) quasiseparable polynomials or simply quasiseparable polynomials. 4. Five diagonal twisted (H ) quasiseparable matrices. The results of Section 2 connect five diagonal CMV matrices (.4) and Fiedler matrices (.6) to the theory of quasiseparable matrices (through the concept of twist transformations). In fact the quasiseparable approach (Section 3) leads to several new results on five diagonal matrices.

11 A quasiseparable approach to five diagonal CMV and companion matrices In this section we investigate recurrence relations satisfied by polynomials {r k (x)} n k=0 related to five diagonal twisted (H ) quasiseparable matrices via r 0 (x) = λ 0 r k (x) = λ 0 λ... λ k det(xi A k k ) k =... n (4.) with λ k 0 being additional parameters. It turns out that in contrast to the case of Hessenberg matrices there are no bijections like (.2) between five diagonal twisted (H ) quasiseparable matrix and polynomial systems. The details will be given in Section 4.2. We start by deriving an entrywise description of five diagonal twisted (H ) - quasiseparable matrices in order to distinguish them from general five diagonal matrices. 4.. Full description of five diagonal twisted (H ) quasiseparable matrices. Consider a 6 6 five diagonal matrix m m 2 m m 2 m 22 m 23 m m A = = 3 m 32 m 33 m 34 m (4.2) 0 0 m 42 m 43 m 44 m 45 m m 53 m 54 m 55 m m 64 m 65 m 66 It is ( ) quasiseparable (Definition.) if and only if all its submatrices A( : i i + n) and A(i + : n : i) for i = 2... n are of rank one. For instance the submatrix A( : 2 3 : 6) highlighted in (4.2) is of rank one if and only if m 3 m 24 = 0. This observation leads to the following simple theorem. Theorem 4.. [Characterizations of five diagonal ( ) quasiseparable matrices] Entrywise characterization. A five diagonal matrix A = [m ij ] n ij= is ( ) quasiseparable if and only if m ii+2 m i+i+3 = m i+2i m i+3i+ = 0 i =... n 3. (4.3) Generator characterization. An ( ) quasiseparable matrix A is five diagonal if and only if it has a choice of generators such that a k a k+ = b k b k+ = 0 k = 2... n 2. (4.4) The conditions (4.3) and (4.4) actually imply that in a five diagonal ( ) quasiseparable matrix every nonzero entry on second sub(super)diagonal is surrounded by two zero entries on that sub(super)diagonal. We give next necessary and sufficient conditions for a five diagonal matrix to be twisted (H ) quasiseparable. Theorem 4.2. A five diagonal matrix A = [m ij ] n ij= is twisted (H ) quasiseparable if and only if m ii+2 m i+i+3 = m i+2i m i+3i+ = 0 i =... n 3 m ii+2 m i+2i = 0 i =... n 2. (4.5) The proof is given in the appendix. The Venn diagram of subclasses of five diagonal matrices is given in Figure 4. and is a consequence of Theorems 4. and 4.2.

12 2 T. Bella V. Olshevsky and P. Zhlobich ( ) quasiseparable matrices condition (4.4) five diagonal matrices twisted (H ) quasiseparable matrices condition (4.5) Fig. 4.. Subclasses of five diagonal matrices 4.2. Non uniqueness of five diagonal twisted (H ) quasiseparable matrices. As we have seen in Section 3. there is a bijection between (H ) quasiseparable matrices together with two nonzero parameters q 0 and q n and systems of polynomials related to them via (4.) with λ k = q k where q k k =... n are generators as in Definition 2.5. In contrast to this bijection for a given system of polynomials (related to some (H ) quasiseparable matrix) there are infinitely many five diagonal twisted (H ) quasiseparable matrices related to it via (4.). We describe next two reasons for this non uniqueness. Reason : Non uniqueness of patterns. Let H be an (H ) quasiseparable matrix with generators {q k d k g k b k h k }. Then the twisted (H ) quasiseparable matrices with patterns ( ) and ( ) obtained from H via twist transformations are five diagonal. For example d g 0 q h 2 d 2 q 2h 3 q 2b 3 q b 2 g 2 d 3 g 3 0 H( ) = 0 q 3h 4 d 4 q 4h 5 q 4b 5 (4.6) q 3 b 4 g 4 d 5 g 5 0 and H( ) = H( ) T. Moreover all five diagonal twisted (H ) quasiseparable matrices obtained from H share the same system of polynomials (Theorem 2.2). Remark 4.3. Five diagonal matrices of the following zero patterns (4.7) 0 0 always satisfy restrictions (4.5) and hence are always twisted (H ) quasiseparable. In addition any matrices of patterns ( ) and ( ) have the first and second of zero patterns of (4.7) respectively.

13 A quasiseparable approach to five diagonal CMV and companion matrices 3 Another fact which is worth mentioning is that if an (H ) quasiseparable matrix H has generators such that b k = 0 for some k then there are two (possibly distinct) five diagonal twisted (H ) quasiseparable matrices of patterns H(i... i k 0 i k+... i n ) and H(i... i k i k+... i n ). Reason 2: Non uniqueness of quasiseparable generators. Let H be an (H ) quasiseparable matrix with generators. It is known that the choice of upper quasiseparable generators {g k b k h k } of H is not unique. On the other hand five diagonal twisted (H ) quasiseparable matrices of fixed patterns obtained from H are defined via these generators in a unique way. For example one can see that different choices of {g k b k h k } lead to different five diagonal matrices (4.6). Example 4.4. Quasiseparable generators (Definition.2) of the companion matrix m m 2 m n m n C = (4.8) were given in Table.3. However it can be easily checked that the generators given in Table 4. also correspond to C. Table 4. Alternative generators of companion matrix k d k a k b k q k g k p k h k k = m k = m 3 0 m 2 k > m k+ 0 If we then apply the twist transformation for odd indices to these alternate generators we arrive at the following five diagonal matrix m m 2 m m 5 F = m (4.9) 0 0 m 7 of pattern ( ) having ( ) quasiseparable generators listed in Table 4.2. Let us note that this new five diagonal companion matrix (4.9) differs from the matrix (.6) derived by Fiedler [40] although it also has the property for which the Fiedler matrix was recognized: the same system of polynomials related via (.) General three term recurrence relations. Let us recall that under the additional restriction of ρ k 0 for each k the corresponding Szegö polynomials satisfy three term recurrence relations φ # k (x) =» µ k x + φ # 0 (x) = µ 0 ρ k ρ k µ k φ # (x) = (x φ # 0 (x) + ρ ρ 0 φ # 0 (x)) µ φ # k (x) ρ k ρ k µ k µ k x φ # k 2 (x) k = 2... n. (4.0)

14 4 T. Bella V. Olshevsky and P. Zhlobich Table 4.2 Quasiseparable generators of matrix (4.9) k d k a k b k q k g k p k h k k = m k = m 3 0 m 2 k > 2 odd 0 m k+ 0 0 k > 2 even 0 0 m k+ 0 This system of polynomials corresponds to a five diagonal twisted (H ) quasiseparable matrix (in fact the CMV matrix). The theorem below gives necessary and sufficient conditions for the existence of general three term recurrence relations for polynomials in terms of five diagonal matrices they are related to via (4.). Theorem 4.5. A system of polynomials R = {r k (x)} n k=0 satisfies three term recurrence relations r 0 (x) = α 0 r (x) = (α x + β ) r 0 (x) r k (x) = (α k x + β k ) r k (x) + (γ k x + δ k ) r k 2 (x) α k 0. (4.) if and only if it is related to a matrix A of the following zero pattern (4.2) with nonzero highlighted entries via (4.) with λ k = α k. Proof. [Necessity] Obviously the matrix A is twisted (H ) quasiseparable (see Remark 4.3) and its general representation is A = d g 0 q h 2 d 2 q 2 h 3 q 2 b 3 q b 2 g 2 d 3 g q 3h 4 d 4 q 4h 5 q 4b 5 q 3 b 4 g 4 d 5 g 5 0. (4.3) Let q k = λ k then all other its generators {d k g k b k h k } are defined uniquely. Moreover the generators h k are all nonzero. Hence there exists a unique (H ) quasiseparable matrix H having generators {q k d k g k b k h k 0} and according to the Theorem 2.2 it is related to the system of polynomials R via (4.). It was proved in Theorem 3. that polynomials related to (H ) quasiseparable matrices with h k 0 satisfy the recurrence relations (4.3) and hence so do {r k (x)} n k=0. [Sufficiency] Let R satisfy three term recurrence relations (4.) it follows from Theorem 3. that there exists a unique (H ) quasiseparable matrix H with q k = α k and h k 0 such that it is related to R via (4.) with λ k = α k. Let A = H( ) be a five diagonal twisted (H ) quasiseparable matrix obtained from H via twist

15 A quasiseparable approach to five diagonal CMV and companion matrices 5 transformations then it has the zero pattern (4.2) and by the Theorem 2.2 is related to the system of polynomials R via (4.) with λ k = α k. The following corollary follows directly from Theorem 3. Remark 4.3 and Theorem 4.5. Corollary 4.6. A system of polynomials satisfies three term recurrence relations (4.) if and only if it is related to a twisted (H ) quasiseparable matrix A. As such the matrix A is twist equivalent to some (not twisted) (H ) quasiseparable matrix B via some pattern (i k ) n k=. Table 3. gives conversion formulas between the three term recurrence relation coefficients and the generators of B. We next present Table 4.3 which gives a conversion from three term recurrence relation coefficients to the generators of the matrix A itself. Table 4.3 Conversion formulas for the generators of a twisted (H ) quasiseparable matrix A in terms of the corresponding three term recurrence relation coefficients. g k b k h k d k if i k = 0 if i k = γ k+d k +δ k α k+ γ k+ α k+ α k β k +γ k α k α k α k 0 α k β k +γ k α k α k p k a k q k if i k = 0 0 α k if i k = γ k+ α k+ γ k+d k +δ k α k+ Example 4.7. Observing a CMV matrix K = ρ 0ρ ρ 0µ 0 µ ρ 2 ρ ρ 2 µ 2 ρ 3 µ 2 µ 3 µ µ 2 ρ µ 2 ρ 2ρ 3 ρ 2µ µ 3 ρ 4 ρ 3ρ 4 µ 4 ρ 5 µ 4 µ 5 µ 3µ 4 ρ 3µ 4 ρ 4ρ 5 ρ 4µ 5 0 (4.4) it is easy to see that the additional restriction ρ k 0 implies that all highlighted entries in (4.4) are not zeros 4 and hence the matrix K satisfies conditions of Theorem 4.5. This proves the existence of recurrence relations (4.0) for polynomials related to K. Example 4.8. Applying Theorem 4.5 to the transpose of a Fiedler matrix F T = m 0 m 2 0 m m 4 0 m (4.5) 4 The definition of the complementary parameters µ k is µ k = j p ρk 2 ρ k < ρ k = which are thus always nonzero.

16 6 T. Bella V. Olshevsky and P. Zhlobich we conclude that three term recurrence relations (4.) must exist for Horner polynomials under the condition m k 0 for k = 2... n. Indeed from the recurrence relations p k (x) = x p k (x) + m k we can get that and hence p k (x) = x p k (x) + m k = = p k x p k 2 m k ( x + m k m k ) p k (x) m k x p k 2. m k 4.4. Szegö type two term recurrence relations. It is well-known that Szegö polynomials related to CMV matrices satisfy two term recurrence relations φ0 (x) φ # 0 (x) = ρ 0 φk (x) µ 0 φ # k (x) = ρ φk k (x) µ k ρ k xφ # k (x). (4.6) In this section we consider the general form of recurrence relations (4.6) (which were called Szegö type in [9]) and derive the class of five diagonal twisted (H ) quasiseparable matrices related to polynomials satisfying them. Theorem 4.9. A system of polynomials R = {r k (x)} n k=0 satisfies Szegö type two term recurrence relations G0 (x) β0 = r 0 (x) δ 0 Gk (x) = r k (x) [ αk β k γ k δ k G k (x) (x + θ k ) r k (x) ] (4.7) with α k δ k β k γ k 0 δ k 0 if and only if it is related to a matrix A of the following zero pattern (4.8) 0 with nonzero highlighted entries via (4.) with λ k = δ k. Proof. [Necessity] The matrix A is twisted (H ) quasiseparable (see Remark 4.3) and its general representation is A = d g 0 q h 2 d 2 q 2h 3 q 2b 3 q b 2 g 2 d 3 g q 3h 4 d 4 q 4h 5 q 4b 5 q 3 b 4 g 4 d 5 g 5 0. (4.9) Fixing q k = λ k then all other generators {d k g k b k h k } are defined uniquely. Moreover the generators b k are all nonzero. Hence there exists a unique (H ) quasiseparable matrix H having generators {q k d k g k b k 0 h k } and according to Theorem

17 A quasiseparable approach to five diagonal CMV and companion matrices it is related to the system of polynomials R via (4.). It is proved in Theorem 3.3 that polynomials related to (H ) quasiseparable matrices with b k 0 satisfy recurrence relations (4.7) and hence so do {r k (x)} n k=0. [Sufficiency] Let R satisfy Szegö type recurrence relations (4.7). Then Theorem 3.3 implies that there exists a unique (H ) quasiseparable matrix H with q k = δ k and b k 0 that is related to R via (4.) with λ k = δ k. Let A = H( ) be a five diagonal twisted (H ) quasiseparable matrix obtained from H via twist transformations which has the zero pattern (4.8) and by Theorem 2.2 is related to the system of polynomials R via (4.) with λ k = δ k. The following corollary follows directly from Theorem 3.3 Remark 4.3 and Theorem 4.9. Corollary 4.0. A system of polynomials satisfies Szegö type recurrence relations (4.7) if and only if it is related to a twisted (H ) quasiseparable matrix A. As such the matrix A is twist equivalent to some (not twisted) (H ) quasiseparable matrix B via some pattern (i k ) n k=. Table 3.2 gives conversion formulas between the Szegö type recurrence relation coefficients and the generators of B. We next present Table 4.4 which gives a conversion from Szegö type recurrence relation coefficients to the generators of the matrix A itself. Table 4.4 Conversion formulas for the generators of a twisted (H ) quasiseparable matrix A in terms of the corresponding Szegö type recurrence relation coefficients. g k b k h k d k if i k = 0 β k (α k δ k β k γ k ) δ k δ k (α k δ k β k γ k ) γ k θ k + β k γ k δ k δ k if i k = δ k 0 θ k + β k γ k δ k δ k p k a k q k if i k = 0 0 if i k = γ k (α k δ k β k γ k ) β k (α k δ k β k γ k ) δ k δ k δ k Let us note also that the transpose of the Fiedler matrix (.6) satisfies the conditions of Theorem 4.9 m 0 m 2 0 m F T = 0 m 4 0 m 5 (4.20) as the highlighted entries in (4.20) are nonzeros. Hence Horner polynomials satisfy Szegö type recurrence relations (4.6). Using the generators from Table.3 and the conversion formulas listed in Table 3.2 we arrive at the following Szegö type recurrence relations for Horner polynomials F0 (x) = p 0 (x) Fk (x) = p k (x) 0 Fk (x). (4.2) m k x p k (x)

18 8 T. Bella V. Olshevsky and P. Zhlobich 4.5. EGO type two term recurrence relations. It was observed in Section 4. that five diagonal twisted (H ) quasiseparable matrices form a proper subclass of ( ) quasiseparable matrices. Hence it is natural to expect that polynomials related to them via (4.) satisfy some special recurrence relations rather than general (.7). The next theorem shows that this is indeed the case. Theorem 4.. A system of polynomials R = {r k (x)} n k=0 satisfies EGO type two term recurrence relations [ F0 (x) 0 Fk (x) βk γ = = k Fk (x) (4.22) r 0 (x) θ0] r k (x) δ k θ k x + ε k r k (x) with θ k = λ k if and only if it is related via (4.) to a five diagonal matrix A = [m ij ] n ij= with entries satisfying m ii+2 m i+i+3 = m i+2i m i+3i+ = 0 i =... n 3 m ii+2 m i+2 = 0 i =... n 2. (4.23) Proof. [Necessity] Let the entries of A satisfy (4.23). Then according to Theorem 4.2 A is twisted (H ) quasiseparable. Hence there exists an (H ) quasiseparable matrix related to the same system of polynomials as A by Theorem 2.2. It immediately follows from Theorem 3.5 that the related polynomials R satisfy the recurrence relations (4.22). [Sufficiency] Let R satisfy EGO type recurrence relations (4.22). Then by Theorem 3.5 there exists an (H ) quasiseparable matrix H that is related to R via (4.). Let A = H( ) be a twisted (H ) quasiseparable matrix obtained from H via twist transformations. Then it is five diagonal (see Remark 4.3) satisfies the conditions (4.23) and is related to the system of polynomials R via (4.) as desired. The following corollary follows directly from Theorem 3.5 and Theorem 4.. Corollary 4.2. A system of polynomials satisfies EGO type recurrence relations (4.22) if and only if it is related to a twisted (H ) quasiseparable matrix A. As such the matrix A is twist equivalent to some (not twisted) (H ) quasiseparable matrix B via some pattern (i k ) n k=. Table 3.3 gives conversion formulas between the EGO type recurrence relation coefficients and the generators of B. We next present Table 4.5 which gives a conversion from EGO type recurrence relation coefficients to the generators of the matrix A itself. Table 4.5 Conversion formulas for the generators of a twisted (H ) quasiseparable matrix A in terms of the corresponding EGO type recurrence relation coefficients. g k b k h k d k if i k = 0 γ k β k δ k θk if i k = ε k θk θ k 0 ε k θk p k a k q k if i k = 0 0 if i k = θ k δ k θ k β k γ k

19 A quasiseparable approach to five diagonal CMV and companion matrices 9 Let us recall that both Fiedler matrices (.6) and CMV matrices (.4) fulfill the conditions of Theorem 4.. Hence Horner and Szegö polynomials must satisfy EGO type recurrence relations (4.22). In particular one can easily check that Horner polynomials satisfy [ F0 (x) 0 = p 0 (x) ] F (x) = p (x) x + m Similarly Szegö polynomials satisfy F0 (x) 0 φ # 0 (x) = µ 0 Fk (x) = p k (x) [ Fk (x) µk ρ k φ # k (x) = µ k [ 0 Fk (x) x p k (x) m k ρ k µ k µ k x ρ k ρ k µ k ]. (4.24) ] [Fk ] (x) φ # k (x). (4.25) 5. Nested factorization of twisted (H ) quasiseparable matrices. In this section we derive a nested factorization of twisted (H ) quasiseparable matrices which we believe might be useful in developing fast algorithms for them. The interpretation of the twist transformation (Definition 2.) can be also given in terms of this new factorization. Theorem 5.. Let H be an arbitrary (H ) quasiseparable matrix specified by its generators as in Definition 2.5 and define the matrices Θ k k by 2 Θ = 4 d g q d 2 I n = 0 n Θ k = I k h k b k q k d k+ I n k Θ n =» In k = 6 4 for k = 2... n n = h n d k d k h k g k d k b k 0 0» 0n d n d n h n where 0 k denotes the k k zero matrix. Then the decomposition 0 n k (5.) H = (... ((Θ Θ )Θ )... )Θ n + n (5.2) holds. The proof of this decomposition is deferred; it will be seen as a special case of Theorem 5.2. Equation (5.2) of this theorem can be viewed as forming the (H ) - quasiseparable matrix H by the iteration H 0 = I n H k = H k Θ k + k k =... n H = H n. (5.3) The next theorem gives the concept of a twist transformation in terms of this decomposition. It states that the effect of an elementary twist transformation at index k changes step k of the decomposition (5.3) via a transpose like operation to that is H k = T k + Θ T k H k ;

20 20 T. Bella V. Olshevsky and P. Zhlobich H k = H k Θ k + k H k = T k + ΘT k H k Twist transformation in terms of decomposition p k a k q k h k b k g k Twist transformation in terms of generators Theorem 5.2. Let H be a twisted (H ) quasiseparable matrix of pattern (i i 2... i n ) with generators {q k d k g k b k h k }. Then it can be constructed by the following procedure: { H k Θ k + k if i k = 0 H 0 = I n H k = T k + ΘT k H k =... n H = H n. (5.4) k if i k = The proof of this theorem is given in the appendix and Theorem 5. follows as a corollary with i k = 0 for k =... n. 6. Appendix. This appendix contains the proofs omitted in the paper. Proof. [Proof of Theorem 4.2] We first prove the only if implication. Let A = [m ij ] be a five diagonal twisted (H ) quasiseparable matrix. Since A is also ( ) quasiseparable it satisfies the conditions of Theorem 4. and hence m ii+2 m i+i+3 = m i+2i m i+3i+ = 0 i =... n 2. (6.) Let A be described by a set of ( ) quasiseparable generators as in the Definition.2. Then its generators satisfy (see Definition 2.8) a i b i = 0 which implies m ii+2 m i+2i = 0 i =... n 2. Therefore the only if implication is proved. Next let A = [m ij ] be a five diagonal matrix satisfying conditions (4.5). These conditions imply that A is ( ) quasiseparable (see Theorem 4.). Hence A has a set of generators as in the Definition.2. Because of the condition (6.) and five diagonallity we can always choose generators to be a i = 0 if m i+2 = 0 and b i = 0 if m ii+2 = 0. Applying twist transformation (Definition 2.) for corresponding indices we can convert matrix A Proof. [Proof of Theorem 5.2] We will show by induction that for every k = 2... n: H k ( : k : k) = H( : k : k) pk =h k = (6.2) In other words every k-th principal submatrix of H k almost equals to the corresponding one of H and equals identically if we modify in the last generators p k and h k. The basis of induction (k = 2) is trivial: [ [ d g H(2 : 2 2 : 2) = h 2 d g H (2 : 2 2 : 2) = for i = 0 H(2 : 2 2 : 2) = ] p 2 q d 2 d q h 2 H p 2 g d (2 : 2 2 : 2) = 2 ] q d 2 d q for i g d =. 2

21 A quasiseparable approach to five diagonal CMV and companion matrices 2 Assume that (6.2) holds for all indices up to k. Consider case i k = 0 the remaining case i k = is essentially the same. Consider matrix H k ( : k + : k + ): 0 H k ( : k : k) h k b k d k d k h k g k d k b k 0 0 q k d k+ (6.3) The last k-th column of matrix H k ( : k : k) equals the last column of matrix H( : k : k) if in the last we set h k =. Therefore performing the matrix product in (6.3) we get: H k ( : k : k ) H k ( : k k)h k H k ( : k k)b k d k h k + (d k d k h k ) d k b k + (g k d k b k ) 0 0 q k d k+ Which is equal to H( : k + : k + ) pk+ =h k+ =. By the induction we get that H n = H pn=h n=. (6.4) Substituting this into recursions (5.4) we get» In H n = H pn =h n = + h n» In H n = H pn =h n = + h n» On» On d n d n h n d n d nh n = H if i n = 0 = H if i n =. REFERENCES [] N.I.Akhiezer The Classical Moment Problem Oliver and Boyd London 965. [2] G. Ammar D. Calvetti and L. Reichel Computing the Poles of Autoregressive Models from the Reflection Coefficients Proceedings of the Thirty-First Annual Allerton Conference on Communication Control and Computing Monticello IL October 993 pp [3] G. Ammar and C. He On an Inverse Eigenvalue Problem for Unitary Hessenberg Matrices Linear Algebra Appl. 28: (995). [4] M. Bakonyi and T. Constantinescu Schur s algorithm and several applications in Pitman Research Notes in Mathematics Series vol. 6 Longman Scientific and Technical Harlow 992. [5] S. Barnett Congenial Matrices Linear Algebra Appl. 4: (98). [6] T. Bella Y. Eidelman I. Gohberg I. Koltracht and V. Olshevsky A Björck-Pereyra-type algorithm for Szegö-Vandermonde matrices based on properties of unitary Hessenberg matrices to appear Linear Algebra and its Applications [7] T. Bella Y. Eidelman I. Gohberg I. Koltracht and V. Olshevsky A fast Björck-Pereyra-like algorithm for solving Hessenberg-quasiseparable-Vandermonde systems submitted. [8] T. Bella Y. Eidelman I. Gohberg V. Olshevsky and E.Tyrtyshnikov Fast inversion of Hessenberg-quasiseparable Vandermonde matrices submitted. [9] T. Bella Y. Eidelman I. Gohberg V. Olshevsky Classifications of three term and two term recurrence relations and digital filter structures via subclasses of quasiseparable matrices submitted to SIAM Journal of Matrix Analysis (SIMAX) [0] T. Bella Y. Eidelman I. Gohberg V. Olshevsky P.Zhlobich Classifications of recurrence relations via subclasses of (Hk)-quasiseparable matrices accepted Numerical Linear Algebra in Signals Systems and Control Springer-Verlag

22 22 T. Bella V. Olshevsky and P. Zhlobich [] Bunse-Gerstner Angelika and He Chun Yang On a Sturm sequence of polynomials for unitary Hessenberg matrices. (English summary) SIAM J. Matrix Anal. Appl. 6 (995) no [2] A.Bruckstein and T.Kailath Some Matrix Factorization Identities for Discrete Inverse Scattering Linear Algebra Appl. 74: (986). [3] A.Bruckstein and T.Kailath Inverse Scattering for Discrete Transmission Line Models SIAM Review 29: (987). [4] A.Bruckstein and T.Kailath An Inverse Scattering Framework for Several Problems in Signal Processing IEEE ASSP Magazine January [5] T.S. Chihara An Introduction to Orthogonal Polynomials Gordon and Breach New York 978. [6] M.J.Cantero L.Moral and L.Velázquez sl five diagonal matrices and zeros of orthogonal polynomials on the unit circle Linear Algebra Appl. 362:29-56 (2003). [7] M.J.Cantero L.Moral and L.Velázquez sl Minimal representations of unitary operators and orthogonal polynomials on the unit circle Linear Algebra Appl. 408:40-65 (2005). [8] M.J.Cantero L.Moral and L.Velázquez sl Measures on the unit circle and unitary truncations of unitary operators Journal of Approximation Theory 39: (2006). [9] Y. Eidelman and I. Gohberg On a new class of structured matrices Integral Equations and Operator Theory 34 (999) [20] Y. Eidelman and I. Gohberg Linear complexity inversion algorithms for a class of structured matrices Integral Equations and Operator Theory 35 (999) [2] Y. Eidelman and I. Gohberg A modification of the Dewilde-van der Veen method for inversion of finitestructured matrices Linear Algebra Appl (2002) [22] Y. Eidelman and I. Gohberg On generators of quasiseparable finite block matrices CAL- COLO 42 (2005) [23] Y. Eidelman I. Gohberg and V. Olshevsky Eigenstructure of Order-One-Quasiseparable Matrices. three term and two term Recurrence Relations Linear Algebra Appl. Volume 405 August 2005 Pages -40. [24] L. Y. Geronimus Polynomials orthogonal on a circle and their applications Amer. Math. Translations 3 p (Russian original 948). [25] G.H. Golub C.F. Van Loan Matrix Computations 3rd ed. The John Hopkins University Press Baltimore CA 996. [26] W. B. Gragg Positive definite Toeplitz matrices the Arnoldi process for isometric operators and Gaussian quadrature on the unit circle (in Russian). In : E.S. Nikolaev (Ed.) Numerical methods in Linear Algebra pp Moskow University Press 982. English translation in : J. Comput. and Appl. Math. 46(993) [27] U. Grenader and G. Szegö Toeplitz forms and Applications University of California Press 958. [28] H. Kimura Generalized Schwarz form and lattice-ladder realization of digital filters IEEE Trans. Circuit & Systems Vol.CAS-32 pp [29] T. Kailath and B. Porat State-space generators for orthogonal polynomials. Prediction theory and harmonic analysis 3 63 North-Holland Amsterdam-New York 983. [30] H.Lev-Ari and T.Kailath Lattice filter parameterization and modeling of nonstationary processes IEEE Trans on Information Theory 30: 2-6 (984). [3] H.Lev-Ari and T.Kailath Triangular factorization of structured Hermitian matrices in Operator Theory : Advances and Applications (I.Gohberg. ed.) vol Birkhäuser Boston 986. [32] J. Maroulas and S. Barnett Polynomials with respect to a general basis. I. Theory J. of Math. Analysis and Appl. 72 (979) [33] V. Olshevsky Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szego-Vandermonde matrices via Discrete Transmission lines. Linear Algebra and Its Applications 285 (998) [34] V. Olshevsky Associated polynomials unitary Hessenberg matrices and fast generalized Parker-Traub and Bjorck-Pereyra algorithms for Szego-Vandermonde matrices invited chapter in the book Structured Matrices: Recent Developments in Theory and Computation (D.Bini E. Tyrtyshnikov P. Yalamov. Eds.) 200 NOVA Science Publ. USA. [35] P. A. Regalia Adaptive IIR filtering in signal processing and control Marcel Dekker New York 995. [36] I. Schur. Uber potenzreihen die in Innern des Einheitskrises Beschrnkt Sind. J. Reine Angew. Math. 47: English translation in: I. Schur Methods in Operator Theory and Signal Processing I. Gohberg ed. Birkhuser