On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations
aa r X i v : . [ m a t h . NA ] J u l On backward errors of structured polynomialeigenproblems solved by structure preserving linearizations
Bibhas Adhikari ∗ and Rafikul Alam † Abstract.
First, we derive explicit computable expressions of structured backward errors of approx-imate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Her-mitian, skew-Hermitian, even and odd polynomials. We also determine minimal structured pertur-bations for which approximate eigenelements are exact eigenelements of the perturbed polynomials.Next, we analyze the effect of structure preserving linearizations of structured matrix polynomialson the structured backward errors of approximate eigenelements. We identify structure preservinglinearizations which have almost no adverse effect on the structured backward errors of approximateeigenelements of the polynomials. Finally, we analyze structured pseudospectra of a structured matrixpolynomial and establish a partial equality between unstructured and structured pseudospectra.
Keywords.
Structured matrix polynomial, structured backward error, pseudospectrum,structured linearization.
AMS subject classifications.
Consider a matrix polynomial P( z ) := P mj =0 z j A j of degree m, where A j ∈ C n × n and A m = 0 . We assume that P is regular, that is, det(P( z )) = 0 for some z ∈ C . We say that λ ∈ C isan eigenvalue of P if det(P( λ )) = 0 . A nonzero vector x ∈ C n (resp., y ∈ C n ) that satisfiesP( λ ) x = 0 (resp., y H P( λ ) = 0) is called a right (resp., left) eigenvector of P corresponding tothe eigenvalue λ. The standard approach to computing eigenelements of P is to convert P intoan equivalent linear polynomial L, called a linearization of P , and employ a numerically back-ward stable algorithm to compute the eigenelements of L , where L( z ) := zX + Y, X ∈ C mn × mn and Y ∈ C mn × mn . It is well known that a matrix polynomial admits several linearizations. Infact, it is shown in [20, 18] that potential linearizations of a matrix polynomial form a vectorspace. Thus choosing an optimal (in some sense) linearization of P is an important first steptowards computing eigenelements of P . In general, a linearization of P can have an adverseeffect on the conditioning of the eigenvalues of P (see, [13]). Hence by analyzing the conditionnumbers of eigenvalues of linearizations, potential linearizations of P have been identified in[13] whose eigenvalues are almost as sensitive to perturbations as that of P . Further, it isshown in [11] that these linearizations are consistent with the backward errors of approximateeigenelements in the sense that they nearly minimize the backward errors.Polynomial eigenvalue problems that occur in many applications possess some distinctivestructures (e.g., Hermitian, even, odd and palindromic) which in turn induce certain spectralsymmetries on the eigenvalues of the matrix polynomials (see, [21, 25, 24, 17, 16] and the ref-erences therein). With a view to preserving spectral symmetry in the computed eigenvalues(and possibly improved accuracy), there has been a lot of interests in developing structuredpreserving algorithms (see, [15, 23, 25, 19] and the references therein). Since linearization isthe standard way to solve a polynomial eigenvalue problem, for a structured matrix polyno- ∗ Department of Mathematics, IIT Guwahati, India, E-mail: [email protected] † Department of Mathematics, IIT Guwahati, India, E-mail: rafi[email protected], rafi[email protected], Fax:+91-361-2690762/2582649. ial it is therefore necessary to choose a structured linearization and then solve the linearproblem by a backward stable structure preserving algorithm. For the accuracy assessmentof computed solution, it is therefore important to understand the sensitivity of eigenvalues ofa structured matrix polynomial with respect to structure preserving perturbations. Also it isequally important to know the structured backward errors of approximate eigenelements of astructured matrix polynomial. Moreover, for a variety of structured polynomials such as sym-metric, skew-symmetric, Hermitian, skew-Hermitian, even, odd and palindromic polynomials,there are infinitely many structured linearizations, see [12, 21]. This poses a genuine problemof choosing one linearization over the other. For computational purposes, it is highly desirableto know how different structured linearizations affect the accuracy of computed eigenelements.Thus the selection of an optimal or a near optimal structured linearization is an importantstep in the solution process of a structured polynomial eigenvalue problem. The sensitivityanalysis of eigenvalues of structured matrix polynomials with respect to structure preservingperturbation has been investigated in [4]. It also provides a recipe for choosing structuredlinearizations whose eigenvalues are almost as sensitive to structure preserving perturbationsas that of the structured matrix polynomials.To complete the investigation, in this paper we analyze structured backward errors ofapproximate eigenelements of symmetric, skew-symmetric, Hermitian, skew-Hermitian, T -even, T -odd, H -even and H -odd polynomials. These structures are defined in Table 1. Themain contribution of this paper is as follows.First, we derive explicit computable expressions for the structured backward errors ofapproximate eigenelements of structured matrix polynomials. We also construct a minimalstructured perturbation so that an approximate eigenelement is the exact eigenelement of thestructured perturbed polynomial. These results generalize similar results in [3] obtained forstructured matrix pencils.Second, we consider structured linearizations that preserve spectral symmetry of a struc-tured matrix polynomial and compare the structured backward errors of approximate eigenele-ments with that of the structured polynomial. For example, a T -even matrix polynomialadmits T -even as well as T -odd linearizations both of which preserve the spectral symmetryof the T -even polynomial. Based on these results we identify structured linearizations whichare optimal in the sense that the structured backward errors of approximate eigenelements ofthe linearizations are bounded above and below by a small constant multiple of that of thestructured polynomials. We show that these linearizations are consistent with the choice oflinearizations discussed in [4] by analyzing structured condition numbers of eigenvalues.Third, we show that the effect of structure preserving linearization on the structuredbackward errors of approximate eigenelements is almost harmless for a wide class of structuredlinearizations. We show that bad effect, if any, of a structure preserving linearization can beneutralized by considering a complementary structured linearization. For example, when Pis a T -even polynomial, we show that any T -even linearization is optimal for eigenvalues λ of P such that | λ | ≤ , and any T -odd linearization is optimal for eigenvalues λ such that | λ | ≥ . In such a case, we show that the backward error of an approximate eigenelement ofthe linearization differ from that of P by no more than a factor of 2 . We show that similarresults hold for other structured polynomials as well. In contrast, it is shown in [4] thatthe condition numbers of eigenvalues of these optimal linearizations differ from that of thepolynomial by a factor of a constant whose size could of the order of the degree of the matrixpolynomial.Finally, we analyze structured pseudospectra of structured matrix polynomials and estab-lish a partial equality between structured and unstructured pseudospectra. Similar study forpalindromic matrix polynomials has been carried out in [2], see also [1, 8].The rest of the paper is organized as follows. In section 2, we review structured poly-nomials and their spectral symmetries. In section 3, we analyze structured backward errorsof approximate eigenpairs of structured polynomials. In section 4, we analyze the effect ofstructure preserving linearizations on the backward errors of approximate eigenelements of2tructure polynomials and provide a recipe for choosing optimal linearizations. Finally, insection 5, we consider structured pseudospectra of structured matrix polynomials.
We consider matrix polynomial of degree m of the form P( z ) := P mj =0 z j A j , where A j ∈ C n × n and A m = 0 . Let P m ( C n × n ) denote the vector space of matrix polynomials of degree at most m. The spectrum of a regular polynomial P ∈ P m ( C n × n ) , denoted by σ (P) , is given by σ (P) := { z ∈ C : det(P( z )) = 0 } . Strictly, speaking σ (P) consists of finite eigenvalues of P . If the leading coefficient of P is singular then P has an infinite eigenvalue. In this paper, weconsider only finite eigenvalues of matrix polynomials. An infinite eigenvalue of P , if any, caneasily be analyzed by considering the reverse polynomial of P (see [6]). We say that ( λ, x, y )is an eigentriple of P if λ is an eigenvalue of P and, x and y are the corresponding nonzeroright and left eigenvectors, that is, P( λ ) x = 0 and y H P( λ ) = 0 . We denote the transpose and conjugate transpose of a matrix A by A T and A H , respec-tively. Define the map P m ( C n × n ) → P m ( C n × n ) , P P ∗ given by P ∗ ( z ) := P mj =0 z j A ∗ j , where A ∗ = A T or A ∗ = A H . The map P P ∗ can be used to define interesting structuredmatrix polynomials such as symmetric, skew-symmetric, Hermitian, skew-Hermitian, ∗ -evenand ∗ -odd matrix polynomials. These structures are defined in Table 1. The table also showsthe eigentriples as well as the spectral symmetries of the eigenvalues, see also [21]. We denotethe set of structured polynomials having one of the structures given in Table 1 by S . Bywriting a pair ( λ, µ ) in the third column of Table 1 we mean that if λ is an eigenvalue of Pthen so is µ. Notice that the eigenvalues of Hermitian and skew-Hermitian polynomials havethe same spectral symmetry. Similarly, the eigenvalues of ∗ -even and ∗ -odd polynomials havethe same spectral symmetry, where ∗ ∈ { T, H } . S Condition spectral symmetry eigentriple symmetric P T ( z ) = P( z ) , ∀ z ∈ C λ ( λ, x, x )skew-symmetric P T ( z ) = − P( z ) , ∀ z ∈ C T -even P T ( z ) = P( − z ) , ∀ z ∈ C ( λ, − λ ) ( λ, x, y ) , ( − λ, y, x ) T -odd P T ( z ) = − P( − z ) , ∀ z ∈ C Hermitian P H ( z ) = P( z ) , ∀ z ∈ C ( λ, λ ) ( λ, x, y ) , ( λ, y, x )skew-hermitian P H ( z ) = − P( z ) , ∀ z ∈ C H -even P H ( z ) = P( − z ) , ∀ z ∈ C ( λ, − λ ) ( λ, x, y ) , ( − λ, y, x ) H -odd P H ( z ) = − P( − z ) , ∀ z ∈ C Table 1: Spectral symmetries of structured polynomials.Let P ∈ S be regular. With a view to obtaining structured backward error of ( λ, x ) ∈ C × C n with x H x = 1 as an approximate eigenpair of P , we now show that there always exists apolynomial △ P ∈ S such that ( λ, x ) is a right eigenpair of P + △ P , that is, (P( λ )+ △ P( λ )) x =0 . Recall that S denotes the set of structured polynomials having one of the structures given inTable 1. In short, we write S ∈ { sym , skew - sym , Herm , skew - Herm , T - even , T - odd , H - even , H - odd } . Theorem 2.1
Let S ∈ { sym , skew - sym , Herm , skew - Herm , T - even , T - odd , H - even , H - odd } and P ∈ S be given by P( z ) = P mj =0 z j A j . Let ( λ, x ) ∈ C × C n be such that x H x = 1 . Set = − P( λ ) x, Λ m := [1 , λ, . . . , λ m ] T and P x := I − xx H . Define △ A j := − xx T A j xx H + λ j k Λ m k [ xr T + rx H − r T x ) xx H ] , if A j = A Tj , − λ j k Λ m k [ xr T − rx H ] , if A j = − A Tj , △ A j := − xx H A j xx H + k Λ m k [ λ j xr H P x + λ j P x rx H ] , if A j = A Hj , − xx H A j xx H − k Λ m k [ λ j xr H P x − λ j P x rx H ] , if A j = − A Hj ,and consider the polynomial △ P( z ) := P mj =0 z j △ A j . Then P( λ ) x + △ P( λ ) x = 0 and △ P ∈ S . Proof:
The proof is computational and is easy to check. (cid:4)
Backward errors of approximate eigenelements of regular matrix polynomials have been sys-tematically analyzed and computable expressions for the backward errors have been derivedby Tisseur in [26] . For our purpose, we require a different norm setup for matrix poly-nomials. We equip P m ( C n × n ) with a norm so that the resulting normed linear space canbe used for perturbation analysis of matrix polynomials. Let P ∈ P m ( C n × n ) be given byP( z ) := P mj =0 A j z j . We define ||| P ||| M := m X j =0 k A j k M / , where k A k M denotes the Frobenius norm when M = F and the spectral norm when M =2 . Accordingly, we say that |||·||| F is the Frobenius norm and |||·||| is the spectral norm on P m ( C n × n ). See [6, 5] for more on norms of matrix polynomials.Let ( λ, x ) ∈ C × C n be such that x H x = 1 and P ∈ P m ( C n × n ) be regular. We denote thebackward error of ( λ, x ) as an approximate eigenelement of P by η M ( λ, x, P) given by η M ( λ, x, P) := inf △ P ∈ P m ( C n × n ) {|||△ P ||| M : P( λ ) x + △ P( λ ) x = 0 } . Setting r := − P( λ ) x and Λ m := [1 , λ, . . . , λ m ] T , it is easily seen that η M ( λ, x, P) = k r k k x k k Λ m k (1)for M = F as well as M = 2 . Indeed, defining △ A j := λ j rx H x H x k Λ m k , j = 0 : m, and consideringthe polynomial △ P( z ) := P mj =0 z j △ A j , we have |||△ P ||| M = k r k / k x k k Λ m k and P( λ ) x + △ P( λ ) x = 0 . Consequently, for simplicity of notation, we denote η M ( λ, x, P) by η ( λ, x, P) . Now suppose that P ∈ S . Then treating ( λ, x ) as an approximate eigenelement of P , wedefine the structured backward error of ( λ, x ) by η S M ( λ, x, P) := inf △ P ∈ S {|||△ P ||| M : P( λ ) x + △ P( λ ) x = 0 } . In view of Theorem 2.1, it follows that η ( λ, x, P) ≤ η S M ( λ, x, P) < ∞ . Structured backwarderrors of approximate eigenelements of structured matrix pencils have been systematicallyanalyzed and computable expressions of the structured backward errors have been derivedin [3]. In this section we generalize these results to the case of structured matrix polynomials.4s we shall see, determining η S ( λ, x, P) is much more difficult than determining η S F ( λ, x, P)and requires solution of norm preserving dilation problem for matrices. The Davis-Kahan-Weinberger solutions of norm preserving dilation problem given below will play an importantrole in the subsequent development. Let
A, B, C and D be matrices of appropriate sizes.Then the following result holds. Theorem 3.1 (Davis-Kahan-Weinberger, [10])
Let
A, B, C satisfy (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) AB (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = µ and (cid:13)(cid:13)(cid:2) A C (cid:3)(cid:13)(cid:13) = µ. Then there exists D such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) A CB D (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = µ. Indeed, those D which havethis property are exactly those of the form D = − KA H L + µ ( I − KK H ) / Z ( I − L H L ) / , where K H := ( µ I − A H A ) − / B H , L := ( µ I − AA H ) − / C and Z is an arbitrary contraction,that is, k Z k ≤ . (cid:4) For a more general version of the above result, see [10].
We now derive structured backward error of ( λ, x ) ∈ C × C n as an approximate eigenpairof symmetric and skew-symmetric matrix polynomials. We also derive minimal structuredperturbations so that ( λ, x ) is an exact eigenpair of the perturbed polynomials. First, weconsider symmetric matrix polynomials. Note that a matrix polynomial P ∈ P m ( C n × n ) issymmetric if and only if all the coefficient matrices of P are symmetric. For a symmetricmatrix polynomial, we have the following result. Theorem 3.2
Let S denote the set of symmetric matrix polynomials in P m ( C n × n ) and let P ∈ S . Let ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m := [1 , λ, . . . , λ m ] T . Then we have η S F ( λ, x, P) = p k r k − | x T r | k Λ m k ≤ √ η ( λ, x, P) and η S ( λ, x, P) = η ( λ, x, P) . Set △ A j := λ j k Λ m k [ xr T + rx H − ( r T x ) xx H ] , j = 0 : m, and consider the polynomial △ P( z ) := P mj =0 z j △ A j . Then △ P is a unique polynomial such that △ P ∈ S , △ P( λ ) x + P( λ ) x = 0 and |||△ P ||| F = η S F ( λ, x, P) . Further, define △ A j := λ j k Λ m k [ xr T + rx H − ( r T x ) xx H ] − λ j x T r P Tx rr T P x k Λ m k ( k r k − | x T r | ) and consider the polynomial △ P( z ) := P mj =0 z j △ A j . Then △ P ∈ S , △ P( λ ) x + P( λ ) x = 0 and |||△ P ||| = η S ( λ, x, P) . Proof:
In view of Theorem 2.1, let △ P ∈ S given by △ P( z ) := P mj =0 △ A j z j be such thatP( λ ) x + △ P( λ ) x = 0 . Let Q ∈ C n × ( n − be such that the matrix Q = [ x Q ] is unitary.Then ] △ A j := Q T △ A j Q = (cid:18) a jj a Tj a j X j (cid:19) , where X j = X Tj is of size n − . Since QQ T = I, we have Q ( △ P( λ )) Q H x = r ⇒ ( △ P( λ )) Q H x = Q T r = (cid:18) x T rQ T r (cid:19) Q H x = e , the first column of the identity matrix, we have (cid:18) P mj =0 λ j a jj P mj =0 λ j a j (cid:19) = (cid:18) x T rQ T r (cid:19) . Hence the minimum norm solutions are a j = λ j Q T r k Λ m k and a jj = λ j x T r k Λ m k , j = 0 : m. Conse-quently, we have g △ A j = λ j x T r k Λ m k ( λ j Q T r k Λ m k ) Tλ j Q T r k Λ m k X j . (2)This shows that the Frobenius norm of ] △ A j is minimized when X j = 0 . Hence we have k△ A j k F = k ] △ A j k F = | a jj | + 2 k a j k . Since Q Q T = I − xx T , we have η S F ( λ, x, P) = s | x T r | k Λ m k + 2 k ( I − xx T ) r k k Λ m k = p k r k − | x T r | k Λ m k . Now from (2), we have △ A j = [ x Q ] λ j x T r k Λ m k ( λ j Q T r k Λ m k ) Tλ j Q T r k Λ m k (cid:18) x H Q H (cid:19) = λ j k Λ m k [ xr T + rx H − ( r T x ) xx H ]which gives the desired polynomial △ P for the Frobenius norm.For the spectral norm, we employ dilation result in Theorem 3.1 to the matrix in (2).Indeed, for µ j := | λ j | k r k k Λ m k , by Theorem 3.1, we have X j = − λ j x T r Q T r ( Q T r ) T k Λ m k ( k r k − | x T r | ) , j = 0 : m, which gives η S ( λ, x, P) = k r k k Λ m k = η ( λ, x, P) . Putting X j in (2) and after simplification wehave △ A j = λ j k Λ m k [ xr T + rx H − ( r T x ) xx H ] − λ j x T r P Tx rr T P x k Λ m k ( k r k − | x T r | )which gives the desired polynomial △ P for the spectral norm. (cid:4)
Remark 3.3 If | x T r | = k r k , then k Q T r k = 0 . Hence considering X j = 0 , j = 0 : m, inthe above proof we obtain the desired results for the spectral norm. Note that in such a casewe have η S F ( λ, x, P) = √ η ( λ, x, P) . Observe that if Y is symmetric and Y x = 0 then Y = P Tx ZP x for some symmetric matrix Z. Consequently, from the proof Theorem 3.2, we have Q j X j Q Hj = P Tx Z j P x , j = 0 : m, forsome symmetric matrices Z j . Hence we have following.
Corollary 3.4
Let P be a symmetric matrix polynomial. For ( λ, x ) ∈ C × C n with x H x = 1 , set r := − P( λ ) x. Then there is a symmetric matrix polynomial Q such that P( λ ) x +Q( λ ) x = 0 if and only if Q( z ) = △ P( z ) + P Tx R( z ) P x for some symmetric polynomial R , where △ P is thesymmetric polynomial given by △ P( z ) := P mj =0 z j △ A j and △ A j := λ j k Λ m k [ xr T + rx H − ( r T x ) xx H ] , j = 0 : m. Next, we consider skew-symmetric matrix polynomials. Note that a matrix polynomialP ∈ P m ( C n × n ) is skew-symmetric if and only if all the coefficient matrices of P are skew-symmetric. For skew-symmetric matrix polynomials we have the following result.6 heorem 3.5 Let S denote the set of skew-symmetric matrix polynomials in P m ( C n × n ) andlet P ∈ S . For ( λ, x ) ∈ C × C n with x H x = 1 , set r := − P( λ ) x. Then we have η S F ( λ, x, P) = √ η ( λ, x, P) , η S ( λ, x, P) = η ( λ, x, P) . For the skew-symmetric polynomial △ P given in Theorem 2.1, we have P( λ ) x + △ P( λ ) x =0 , |||△ P ||| F = η S F ( λ, x, P) and |||△ P ||| = η S ( λ, x, P) . Proof:
The proof is the same as that of Theorem 3.2 except that △ A j is skew-symmetric for j = 0 : m. This gives ] △ A j = − ( λ j Q T r k Λ m k ) Tλ j Q T r k Λ m k X j . (3)Setting X j = 0 , we obtain the results for the Frobenius norm.Setting µ j := | λ j | k r k k Λ m k and invoking Theorem 3.1, it is easily seen that the spectral normof ] △ A j in (3) is minimized when X j = 0 . Hence the desired results follow for the spectralnorm. (cid:4)
Note that if Y is a skew-symmetric matrix and Y x = 0 then Y = P Tx ZP x for someskew-symmetric matrix Z. Hence we have the following result.
Corollary 3.6
Let P ∈ P m ( C n × n ) be a skew-symmetric matrix polynomial. For ( λ, x ) ∈ C × C n with x H x = 1 , set r := − P( λ ) x. Then there is a skew-symmetric matrix polynomial Q suchthat P( λ ) x + Q( λ ) x = 0 if and only if Q( z ) = △ P( z ) + P Tx R( z ) P x for some skew-symmetricpolynomial R , where △ P is the skew-symmetric polynomial given by △ P( z ) := P mj =0 z i △ A j and △ A j = − λ j k Λ m k [ xr T − rx H ] , j = 0 : m. For backward perturbation analysis of T -even and T -odd polynomials, we need the even indexprojection Π e : C m +1 → C m +1 given byΠ e ([ x , x , . . . , x m − , x m ] T ) := (cid:26) [ x , , x , , . . . , x m − , , x m ] T , if m is even , [ x , , x , , . . . , , x m − , T , if m is odd.Note that “0” is considered as even number. Observe that I − Π e is the odd index projection.Recall that a matrix polynomial P ∈ P m ( C n × n ) given by P( z ) := P mj =0 A j z j is T -even ifand only if A j is symmetric when j is even (including j = 0) and A j is skew-symmetric when j is odd. We have the following result for T -even matrix polynomials. Theorem 3.7
Let S denote the set of T -even matrix polynomials in P m ( C n × n ) . Let P ∈ S and ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m :=[1 , λ, . . . , λ m ] T . Then we have η S F ( λ, x, P) = s | x T r | k Π e (Λ m ) k + 2 k r k − | x T r | k Λ m k , η S ( λ, x, P) = s | x T r | k Π e (Λ m ) k + k r k − | x T r | k Λ m k . In particular, if m is odd and | λ | = 1 then we have η S F ( λ, x, P) = √ η ( λ, x, P) and η S ( λ, x, P) = η ( λ, x, P) . For j = 0 : m, define E j := λ j k Π e (Λ m ) k ( x T r ) xx H + λ j k Λ m k [ xr T P x + P Tx rx H ] , if j is even, λ j k Λ m k [ P Tx rx H − xr T P x ] , if j is odd . hen △ P( z ) := P mj =0 z j E j is a unique T -even polynomial in S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| F = η S F ( λ, x, P) . Further, for j = 0 : m, defining △ A j := E j − λ j x T r P Tx rr T P x k Π e (Λ m ) k ( k r k − | x T r | ) , if j is even, E j , if j is odd,we obtain a T-even polynomial △ P( z ) := P mj =0 z j △ A j in S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| = η S ( λ, x, P) . Proof:
In view of Theorem 2.1, let △ P ∈ S be such that P( λ ) x + △ P( λ ) x = 0 . Assum-ing that △ P is given by △ P( z ) := P mj =0 △ A j z j , and arguing similarly as in the proofs ofTheorems 3.2 and 3.5, we have ] △ A j = (cid:18) a jj a Tj a j X j (cid:19) , X Tj = X j when j is even, and ] △ A j = (cid:18) b Tj − b j Y j (cid:19) , Y Tj = − Y j when j is odd. Consequently, we have (cid:18) P j λ j a jj P j -even λ j a j − P j -odd λ j b j (cid:19) = (cid:18) x T rQ T r (cid:19) . Hence the smallest norm solutions are a jj = λ j k Π e (Λ m ) k x T r, a j = λ j k Λ m k Q T r, b j = − λ j k Λ m k Q T r. Therefore, we have ] △ A j = Q T △ A j Q = λ j k Π e (Λ m ) k x T r ( λ j Q T r k Λ m k ) Tλ j Q T r k Λ m k X j , if j is even − ( λ j Q T r k Λ m k ) Tλ j Q T r k Λ m k Y j , if j is odd. (4)Setting X j = 0 = Y j and using the fact that Q Q T = I − xx T , we obtain the desired unique T -even polynomial △ P( z ) := P mj =0 z j E j such that |||△ P ||| F = η S F ( λ, x, P) = s | x T r | k Π e (Λ m ) k + 2 k r k − | x T r | k Λ m k . When m is odd and | λ | = 1 , it is easily seen that k Π e (Λ m ) k = k Λ m k . Hence we have η S F ( λ, x, P) = √ η ( λ, x, P) . For the spectral norm, setting µ j := q | λ j | | x T r | k Π e (Λ m ) k + | λ j | ( k r k −| x T r | ) k Λ m k when j is even, and µ j := q | λ j | ( k r k −| x T r | ) k Λ m k when j is odd, and applying Theorem 3.1 to the matrices in (4),we have X j = − λ j x T r Q T r ( Q T r ) T k Π e (Λ m ) k ( k r k − | x T r | ) and Y j = 0 . Consequently, we have η S ( λ, x, P) = s | x T r | k Π e (Λ m ) k + k r k − | x T r | k Λ m k . From (4), we have △ A j = λ j x T rxx H k Π e (Λ m ) k + λ j k λ k [ xr T P x + P Tx rx H ] + Q X j Q H , if j is even λ j k λ k [ P Tx rx H − xr T P x ] + Q Y j Q H , if j is odd.8ubstituting X j and Y j in △ A j we obtain the desired T -even matrix polynomial △ P for thespectral norm. (cid:4)
Remark 3.8 If | x T r | = k r k then k Q T r k = 0 . Hence considering X j = 0 = Y j in the aboveproof, we obtain the desired result for the spectral norm. Note that in such a case we have η S F ( λ, x, P) = √ η S ( λ, x, P) = √ η ( λ, x, P) . Recall that when A is symmetric (resp., skew-symmetric) and Ax = 0 then A = P Tx ZP x forsome symmetric (resp., skew-symmetric) matrix Z. Consequently, from the proof Theorem 3.7it follows that △ A j := E j + P Tx Z j P x , where Z j = Z Tj when j is even, and Z Tj = − Z j when j is odd. Hence we have the following result. Corollary 3.9
Let P be a T -even matrix polynomial in P m ( C n × n ) . Let ( λ, x ) ∈ C × C n besuch that x H x = 1 . Then there is a T -even matrix polynomial Q such that P( λ ) x + Q( λ ) x = 0 if and only if Q( z ) = △ P( z ) + P Tx R( z ) P x for some T -even matrix polynomial R ∈ P m ( C n × n ) , where △ P( z ) := P mj =0 E j z j and E j ’s are given in Theorem 3.7. Next, we consider backward error of T -odd polynomials. Observe that a matrix polynomialP ∈ P m ( C n × n ) given by P( z ) := P mj =0 A j z j is T -odd if and only if A j is skew-symmetric when j is even (including j = 0) and A j is symmetric when j is odd. Theorem 3.10
Let S denote the set of T -odd matrix polynomials in P m ( C n × n ) . Let P ∈ S and ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m :=[1 , λ, . . . , λ m ] T . Then we have η S F ( λ, x, P) = ( q | x T r | k ( I − Π e )(Λ m ) k + 2 k r k −| x T r | k Λ m k , if λ = 0 , √ η ( λ, x, P) , if λ = 0 ,η S ( λ, x, P) = ( q | x T r | k ( I − Π e )(Λ m ) k + k r k −| x T r | k Λ m k , if λ = 0 ,η ( λ, x, P) , if λ = 0 . In particular, if m is odd and | λ | = 1 we have η S F ( λ, x, P) = √ η ( λ, x, P) and η S ( λ, x, P) = η ( λ, x, P) . For j = 0 : m, define F j := λ j k Λ m k [ P Tx rx H − xr T P x ] , if j is even λ j xx T rx H k ( I − Π e )(Λ m ) k + λ j k Λ m k [ xr T P x + P Tx rx H ] , if j is odd.Then △ P( z ) := P mj =0 z j F j is a unique T -odd polynomial in S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| F = η S F ( λ, x, P) . Further, for j = 0 : m, define △ A j := F j when j is even, and △ A j := F j − λ j x T rP Tx rr T P x k ( I − Π e )(Λ m ) k ( k r k − | x T r | ) when j is odd. Then △ P( z ) := P mj =0 z j △ A j is a T -odd polynomial in S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| = η S ( λ, x, P) . Proof:
The desired results follow from the proof of Theorem 3.7 by interchanging the role of △ A j for even j and odd j. (cid:4) We have the following results whose proof is immediate.
Corollary 3.11
Let P be a T -odd matrix polynomial in P m ( C n × n ) . Let ( λ, x ) ∈ C × C n besuch that x H x = 1 . Then there is a T -odd matrix polynomial Q such that P( λ ) x + Q( λ ) x = 0 if and only if Q( z ) = △ P( z ) + P Tx R( z ) P x for some T -odd matrix polynomial R ∈ P m ( C n × n ) , where △ P( z ) := P mj =0 F j z j and F j ’s are given in Theorem 3.10. .3 Hermitian and skew-Hermitian matrix polynomials We now consider structured backward errors of approximate eigenelements of Hermitian andskew-Hermitian matrix polynomials. We proceed as follows. Let S ⊂ P m ( C n × n ) and ω ∈ C be such that | ω | = 1 . We set S ω := { ω P : P ∈ S } . Then for P ∈ P m ( C n × n ) , it is easily seenthat η S F ( λ, x, P) = η S ω F ( λ, x, ω P) and η S ( λ, x, P) = η S ω ( λ, x, ω P) . (5)Note that a matrix polynomial P ∈ P m ( C n × n ) is Hermitian (resp., skew-Hermitian) if andonly if all the coefficient matrices of P are Hermitian (resp., skew-Hermitian). Let Herm and skew - Herm , respectively, denote the set of Hermitian and skew-Hermitian matrix polynomialsin P m ( C n × n ) . Then noting that a matrix X ∈ C n × n is Hermitian if and only if iX is skew-Hermitian, it easily seen that the maps Herm −→ skew - Herm , P i P and skew - Herm −→ Herm , Q i Q (6)are isometric isomorphisms. Thus, in view of (5) and (6), it follows that the structuredbackward error of ( λ, x ) as an approximate eigenpair of a skew-Hermitian polynomial canbe obtained from the structured backward error of ( λ, x ) as an approximate eigenpair ofa Hermitian matrix polynomial and vice-versa. We therefore analyze structured backwardperturbation of Hermitian matrix polynomials.For x ∈ C n , we denote by Re ( x ) and Im ( x ) , respectively, the real and the imaginary partsof x. Then we have x = Re ( x ) + i Im ( x ) . We denote the real and imaginary part of a complexnumber z ∈ C by re ( z ) and im ( z ) , respectively. We denote the Moore-Penrose pseudo-inverseof A by A † and the canonical basis of C m +1 by e j , j = 0 : m. Theorem 3.12
Let
Herm denote the set of Hermitian matrix polynomials in P m ( C n × n ) . Let P ∈ Herm and ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m := [1 , λ, . . . , λ m ] T . Then we have η Herm F ( λ, x, P) = √ k r k −| x H r | k Λ m k ≤ √ η ( λ, x, P) , if λ ∈ R , q k b r k + k r k −| x H r | ) k Λ m k , if λ ∈ C \ R ,η Herm ( λ, x, P) = ( η ( λ, x, P) , if λ ∈ R , q k b r k + k r k −| x H r | k Λ m k , if λ ∈ C \ R , where b r := (cid:20) Re (Λ m ) T Im (Λ m ) T (cid:21) † (cid:20) re ( x H r ) im ( x H r ) (cid:21) . For the Frobenius norm, define △ A j := λ j k Λ m k ( xr H + rx H − ( r H x ) xx H ) , when λ ∈ R ,e Tj b rxx H + 1 k Λ m k [ λ j P x rx H + λ j xr H P x ] , when λ ∈ C \ R . Then △ P( z ) := P mj =0 z j △ A j is a unique Hermitian polynomial in Herm such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| F = η Herm F ( λ, x, P) . For the spectral norm, define △ A j := λ j k Λ m k ( rx H + xr H − ( r H x ) xx H ) − λ j x H rP x rr H P x k Λ m k ( k r k − | x H r | ) , when λ ∈ R ,e Tj b rxx H + 1 k Λ m k [ λ j P x rx H + λ j xr H P x ] − e Tj b r P x rr H P x k r k − | x H r | , when λ ∈ C \ R . Then △ P( z ) := P mj =0 z j △ A j is a Hermitian polynomial in Herm such that P( λ ) x + △ P( λ ) x =0 and |||△ P ||| = η Herm ( λ, x, P) . roof: Again, in view of Theorem 2.1, let △ P( z ) = P mj =0 z j △ A j be a Hermitian polynomialsuch that △ P( λ ) x + P( λ ) x = 0 . Choosing a unitary matrix Q := [ x, Q ] , we have ] △ A j := Q H △ A j Q = (cid:18) a jj a Hj a j X j (cid:19) , Q H r = (cid:18) x H rQ H r (cid:19) . Now △ P( λ ) x + P( λ ) x = 0 ⇒ (cid:18) P mj =0 λ j a jj P mj =0 λ j a j (cid:19) = (cid:18) x H rQ H r (cid:19) . The minimum norm solution of P mj =0 λ j a j = Q H r is given by a j = λ j k Λ m k Q H r. Now suppose that λ ∈ R . Then the minimum norm solution of P mj =0 λ j a jj = x H r is givenby a jj = λ j k Λ m k x H r ∈ R . Hence for λ ∈ R , we have ] △ A j = λ j k Λ m k x H r ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r X j ! , j = 0 : m. (7)For the Frobenius norm, setting X j = 0 we obtain η Herm F ( λ, x, P) = √ k r k −| r H x | k Λ m k and thedesired Hermitian polynomial △ P . For the spectral norm, setting µ j := | λ j | k r k k Λ m k and applying Theorem 3.1 to (7), we obtain X j = − λ j x H r ( Q H r )( Q H r ) H k Λ m k ( k r k − | x H r | ) . This gives η Herm ( λ, x, P) = k r k k Λ m k = η ( λ, x, P) . Now substituting X j in (7) and simplifyingthe expression, we obtain the desired Hermitian polynomial △ P . Next, suppose that λ ∈ C \ R . Then the minimum norm solution of P mj =0 λ j a jj = x H r isobtained by solving (cid:18) P mj =0 re ( λ j ) a jj P mj =0 im ( λ j ) a jj (cid:19) = (cid:18) re ( x H r ) im ( x H r ) (cid:19) ⇒ a ... a mm = (cid:18) Re (Λ m ) T Im (Λ m ) T (cid:19) † (cid:18) re ( x H r ) im ( x H r ) (cid:19) =: b r. Therefore we have a jj = e Tj b r. Hence for λ ∈ C \ R , we have Q H △ A j Q = e Tj b r ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r X j ! , j = 0 : m. (8)Thus, for the Frobenius norm, setting X j = 0 we obtain η Herm F ( λ, x, P) = s k b r k + 2 k r k − | r H x | k Λ m k and the desired Hermitian polynomial △ P . For the spectral norm, setting µ j := q | e Tj b r | + | λ j | ( k r k −| x H r | ) k Λ m k and applying Theo-rem 3.1 to the matrix in (8), we have X j = − e Tj b r ( Q H r )( Q H r ) H k r k − | x H r | , j = 0 : m. This gives η Herm ( λ, x, P) = s k b r k + k r k − | x H r | k Λ m k . X j in (8) and simplifying the expression, we have △ A j = e Tj b rxx H + 1 k Λ m k [ λ j P x rx H + λ j xr H P x ] − e Tj b r P x rr H P x k r k − | x H r | . Hence the results follow. (cid:4)
Remark 3.13 If | x H r | = k r k then k Q H r k = 0 . Hence considering X j = 0 , j = 0 : m, weobtain the desired results for the spectral norm. Let x ∈ C n be such that x H x = 1 . If A ∈ C n × n is Hermitian and Ax = 0 then it is easilyseen that A = ( I − xx H ) Z ( I − xx H ) for some Hermitian matrix Z. Consequently, in viewof Theorem 3.12, we have an analogue of the result in Corollary 3.4 for Hermitian matrixpolynomials.Note that, in view of (5) and (6), structured backward error of ( λ, x ) as an approximateeigenpair of a skew-Hermitian matrix polynomial follows from Theorem 3.12. Indeed, letQ ∈ skew - Herm ⊂ P m ( C n × n ) be a skew-Hermitian matrix polynomial. Then P := i Q ∈ Herm ⊂ P m ( C n × n ) . Hence by (5) and (6), we have η skew - Herm M ( λ, x, Q) = η Herm M ( λ, x, P) . Now,let △ P be the matrix polynomial given in Theorem 3.12 such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| M = η Herm M ( λ, x, P) . Then setting △ Q := − i △ P , we have △ Q ∈ skew - Herm such thatQ( λ ) x + △ Q( λ ) x = 0 and |||△ Q ||| M = η skew - Herm M ( λ, x, Q) . We now derive structured backward errors of approximate eigenelements of H -even and H -odd matrix polynomials. Recall that a matrix polynomial P ∈ P m ( C n × n ) given by P( z ) := P mj =0 A j z j is H -even if and only if A j is Hermitian when j is even (including j = 0) and A j is skew-Hermitian when j is odd. Let H - even and H - odd , respectively, denote the setof H -even and H -odd matrix polynomials in P m ( C n × n ) . Then, as in the case of Hermitianmatrix polynomials in (6), it is easily seen that the map H - even −→ H - odd , P i P and H - odd −→ H - even , Q i Q (9)are isometric isomorphisms. Consequently, we only need to prove the results either for H - even or for H - odd matrix polynomials. Recall that A † is the Moore-Penrose pseudo-inverse of A and e j , j = 0 : m, is the canonical basis of C m +1 . Theorem 3.14
Set S := H - even ⊂ P m ( C n × n ) . Let P ∈ S and ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m := [1 , λ, . . . , λ m ] T . Then we have η S F ( λ, x, P) = √ k r k −| x H r | k Λ m k ≤ √ η ( λ, x, P) , if λ ∈ i R , q k b r k + k r k −| x H r | ) k Λ m k , if λ ∈ C \ i R ,η S ( λ, x, P) = η ( λ, x, P) , if λ ∈ i R , q k b r k + k r k −| x H r | k Λ m k , if λ ∈ C \ i R , where b r := (cid:20) Π e Re (Λ m ) T − ( I − Π e ) Im (Λ m ) T Π e Im (Λ m ) T + ( I − Π e ) Re (Λ m ) T (cid:21) † (cid:20) re ( x H r ) im ( x H r ) (cid:21) . For j = 0 : m, set E j := 1 k Λ m k [ λ j P x rx H + λ j xr H P x ] and F j := 1 k Λ m k [ λ j P x rx H − λ j xr H P x ] . or the Frobenius norm, define △ A j := λ j k Λ m k [ xr H + rx H − ( r H x ) xx H ] when λ ∈ i R , and △ A j := (cid:26) e Tj b rxx H + E j , if j is even, ie Tj b rxx H + F j , if j is odd,when λ ∈ C \ i R , for j = 0 : m. Then △ P( z ) := P mj =0 z j △ A j is a unique H -even polynomialin S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| F = η S F ( λ, x, P) . For the spectral norm, define △ A j := λ j k Λ m k [ xr H + rx H − ( r H x ) xx H ] + ( − j +1 λ j x H rP x rr H P x k Λ m k ( k r k − | x H r | ) when λ ∈ i R , and △ A j := e Tj b rxx H + E j + ( − j +1 e Tj b r P x rr H P x k r k − | x H r | , if j is even, ie Tj b rxx H + F j − i ( − j +1 e Tj b r P x rr H P x k r k − | x H r | , if j is odd,when λ ∈ C \ i R . Then △ P( z ) := P mj =0 z j △ A j is an H -even polynomial in S such that P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| = η S ( λ, x, P) . Proof:
By Theorem 2.1 there exists an H-even matrix polynomial △ P( z ) = P mj =0 z j △ A j such that △ P( λ ) = r. Now choosing a unitary matrix Q := [ x, Q ] , we have △ A j = Q (cid:18) a jj a Hj a j X j (cid:19) Q H , X Hj = X j if j is even, and △ A j = Q (cid:18) ia jj a Hj − a j Y j (cid:19) Q H , Y Hj = − Y j if j is odd. Notice that a jj is real for all j. Then △ P( λ ) x = r gives (cid:18) P j -even λ j a jj + i P j -odd λ j a jj P j -even λ j a j − P j -odd λ j a j (cid:19) = (cid:18) x H rQ H r (cid:19) . The mini-mum norm solution of P j -even λ j a j − P j -odd λ j a j = Q H r is given by a j = λ j k Λ m k Q H r if j is even, and a j = − λ j k Λ m k Q H r if j is odd.Now suppose that λ ∈ i R . Then the minimum norm solution for a jj is given by a jj = λ j k Λ m k x H r when j is even, and a jj = − i λ j k Λ m k x H r when j is odd. Hence a jj ∈ R when j iseven, and ia jj ∈ i R when j is odd. Consequently, we have Q H △ A j Q = λ j k Λ m k x H r ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r X j (10)when j is even, and Q H △ A j Q = λ j k Λ m k x H r − ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r Y j (11)when j is odd. Setting X j = 0 = Y j in (10) and (11), we obtain η S F ( λ, x, P) = √ k r k −| r H x | k Λ m k and the desired △ A j . Next, suppose that λ ∈ C \ i R . Then P j -even λ j a jj + i P j -odd λ j a jj = x H r gives (cid:18) P j -even re ( λ j ) a jj − P j -odd im ( λ j ) a jj P j -even im ( λ j ) a jj + P j -odd re ( λ j ) a jj (cid:19) = (cid:18) re ( x H r ) im ( x H r ) (cid:19) . a a ... a mm = (cid:18) Π e Re (Λ m ) T − ( I − Π e ) Im (Λ m ) T Π e Im (Λ m ) T + ( I − Π e ) Re (Λ m ) T (cid:19) † (cid:18) re ( x H r ) im ( x H r ) (cid:19) = b r ⇒ a jj = e Tj b r. Consequently, we have Q H △ A j Q = e Tj b r ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r X j (12)when j is even, and Q H △ A j Q = ie Tj b r − ( λ j k Λ m k Q H r ) Hλ j k Λ m k Q H r Y j (13)when j is odd. Now setting X j = 0 = Y j in (12) and (13), we have the desired matrices △ A j , j = 0 : m, and η S F ( λ, x, P) = s k b r k + 2 k r k − | x H r | k Λ m k . This completes the proof for theFrobenius norm.For the spectral norm, consider µ j := | λ j | k r k k Λ m k when λ ∈ i R . Then applying Theorem 3.1to the matrices in (10) and (11), we obtain X j = − λ j x H r ( Q H r )( Q H r ) H k Λ m k ( k r k − | x H r | ) and Y j = λ j x H r ( Q H r )( Q H r ) H k Λ m k ( k r k − | x H r | ) . This gives η S ( λ, x, P) = k r k k Λ m k . Now substituting X j and Y j in (10) and (11), we obtain thedesired matrices △ A j , j = 0 : m. When λ ∈ C \ i R , considering µ j := q | e Tj b r | + | λ j | ( k r k −| x H r | ) k Λ m k and applying Theo-rem 3.1 to the matrices in (12) and (13), we obtain X j = − e Tj b r ( Q H r )( Q H r ) H k r k − | x H r | and Y j = − ie Tj b r ( Q H r )( Q H r ) H k r k − | x H r | . Consequently, we have η S ( λ, x, P) = s k b r k + k r k − | x H r | k Λ m k . Substituting X j and Y j in (12) and (13), we obtain the desired matrices △ A j , j = 0 : m. (cid:4) Let x ∈ C be such that x H x = 1 . If X ∈ C n × n is skew-Hermitian and Xx = 0 then it iseasily seen that X = ( I − xx H ) Z ( I − xx H ) for some skew-Hermitian matrix Z. Consequently,it follows that an analogue of the result in Corollary 3.11 holds for H -even matrix polynomials.Observe that, in view of (5) and (9), the structured backward error of ( λ, x ) as anapproximate eigenpair of an H -odd matrix polynomial follows from Theorem 3.14. In-deed, let Q be an H -odd matrix polynomial in P m ( C n × n ) . Set S e := H - even ⊂ P m ( C n × n )and S o := H - odd ⊂ P m ( C n × n ) . Then P := i Q ∈ S e . Hence by (5) and (9), we have η S o M ( λ, x, Q) = η S e M ( λ, x, P) . Now, let △ P be the matrix polynomial given in Theorem 3.14 suchthat △ P ∈ S e , P( λ ) x + △ P( λ ) x = 0 and |||△ P ||| M = η S e M ( λ, x, P) . Then setting △ Q := − i △ P , we have △ Q ∈ S o , Q( λ ) x + △ Q( λ ) x = 0 and |||△ Q ||| M = η S o M ( λ, x, P) . .5 Polynomials with coefficients in Lie and Jordan algebras We mention that the structured backward perturbation analysis of structured matrix polyno-mials discussed so far can easily be extended to more general structured matrix polynomials inwhich the coefficient matrices are elements of appropriate Jordan and/or Lie algebras. Indeed,let M be a unitary matrix such that M T = M or M T = − M. Consider the Jordan algebra J := { A ∈ C n × n : M − A T M = A } and the Lie algebra L := { A ∈ C n × n : M − A T M = − A } associated with the scalar product ( x, y ) y T M x.
Consider a polynomial P( z ) := P mj =0 z j A j . Then by imposing the condition that the polynomial M P given by M P( z ) = P mj =0 λ j M A j iseither symmetric or skew-symmetric or T -even or T -odd, we obtain various structured matrixpolynomials. Said differently, S ⊂ P m ( C n × n ) defines a class of structured matrix polynomialsif M S ∈ { sym , skew - sym , T - even , T - odd } . Hence if P ∈ S then the results obtained in theprevious section are easily extended to P by replacing A j and r := − P( λ ) x by M A j and M r, respectively.Similarly, when M is unitary and M = M H or M = − M H , we consider the Jordanalgebra J := { A ∈ C n × n : M − A H M = A } and the Lie algebra L := { A ∈ C n × n : M − A H M = − A } associated with the scalar product ( x, y ) y H M x.
Then a class ofstructured matrix polynomials S ⊂ P m ( C n × n ) is obtained by imposing the condition that M S ∈ { Herm , skew - Herm , H - even , H - odd } . Hence the results obtained in the previous sectionare easily extended to P ∈ S by replacing A j and r := − P( λ ) x by M A j and M r, respectively.In particular, when M := J, where J := (cid:18) I − I (cid:19) ∈ C n × n , the Jordan algebra J consistsof skew-Hamiltonian matrices and the Lie algebra L consists of Hamiltonian matrices. So, forexample, considering the polynomial P( z ) := P mj =0 z j A j , where A j s are Hamiltonian when j is even and skew-Hamiltonian when j is odd, we see that the polynomial J P( z ) = P mj =0 z j JA j is H -even. Hence extending the results obtained for H -even polynomial to the case of P , wehave the following. Theorem 3.15
Let S denote set of polynomials of the form P( z ) = P mj =0 z j A j where A j is Hamiltonian when j is even, and A j is skew-Hamiltonian when j is odd. Let P ∈ S and ( λ, x ) ∈ C × C n be such that x H x = 1 . Set r := − P( λ ) x, P x := I − xx H and Λ m :=[1 , λ, . . . , λ m ] T . Then we have η S F ( λ, x, P) = √ k r k −| x H Jr | k Λ m k ≤ √ η ( λ, x, P) , if λ ∈ i R , q k b r k + k r k −| x H Jr | ) k Λ m k , if λ ∈ C \ i R ,η S ( λ, x, P) = η ( λ, x, P) , if λ ∈ i R , q k b r k + k r k −| x H Jr | k Λ m k , if λ ∈ C \ i R , where b r := (cid:20) Π e ( Re (Λ m ) T ) − ( I − Π e )( Im (Λ m ) T )Π e ( Im (Λ m ) T ) + ( I − Π e )( Re (Λ m ) T ) (cid:21) † (cid:20) re ( x H Jr ) im ( x H Jr ) (cid:21) . As we have mentioned before, linearization is the standard approach to solving a polynomialeigenvalue value problem. It is well known that important classes of structured matrix poly-nomials admit structured linearizations [18, 12, 21, 20]. However, the process of linearizing amatrix polynomial (structure preserving or not) has its side effect too. It increases the sensi-tivity of eigenvalues of the matrix polynomial (see, [13, 1, 4]). Therefore, it is important toidentify linearizations whose eigenelements are almost as sensitive to perturbations as those ofthe matrix polynomial. Obviously, condition numbers of eigenvalues and backward errors ofapproximate eigenelements have an important role to play in identifying such linearizations.15or an unstructured polynomial P ∈ P m ( C n × n ) , Higham et al. [13, 11] provide a recipe forchoosing a linearization by analyzing condition numbers of eigenvalues and backward errorsof approximate eigenpairs. For structured matrix polynomials, a recipe for choosing a struc-tured linearization has been provided in [4] by analyzing structured condition numbers ofeigenvalues. With a view to identifying optimal and near optimal structured linearizationsof a structured matrix polynomials, in this section, we analyze the influence of structuredlinearizations on the structured backward errors of approximate eigenelements. It turns outthat linearizations which minimize structured backward errors also minimize the structuredcondition numbers. Therefore our results are consistent with those in [4]. We thus provide arecipe for choosing structured linearizations of a structured matrix polynomial which minimizestructured backward errors as well as the structured condition numbers.For a ready reference, we briefly review some basic results about linearizations of P ∈ P m ( C n × n ) , for details, see [20, 18, 12, 21]. For our purpose, it is enough to consider thevector space L (P) given by [20] L (P) := { L( λ ) : L( λ ) . (Λ m − ⊗ I n ) = v ⊗ P( λ ) , v ∈ C m } , where Λ m − := [ λ m − , λ m − , . . . , T , ⊗ is the Kronecker product and v is called the rightansatz vector for L . Let v := [ v , v , . . . , v m ] T ∈ C m be a right ansatz vector. Then thescalar polynomial p ( x ; v ) := v x m − + v x m − + . . . + v m − x + v m is referred to as the “ v -polynomial” of the vector v, see [18, 20]. The convention is that p ( x ; v ) is said to have a rootat ∞ whenever v = 0 . Let L( λ ) = λX + Y ∈ L (P) be a linearization of P corresponding tothe right ansatz vector v ∈ C m . Then for x ∈ C n , the following holds k L( λ )(Λ m − ⊗ x ) k = k v k k P( λ ) x k , (14) | (Λ m − ⊗ x ) T L( λ )(Λ m − ⊗ x ) | = | Λ Tm − v | | x T P( λ ) x | , (15) | (Λ m − ⊗ x ) H L( λ )(Λ m − ⊗ x ) | = | Λ Hm − v | | x H P( λ ) x | . (16)Observe from (14) that ( λ, x ) is an eigenelement of P if and only if ( λ, Λ m − ⊗ x ) is aneigenelement of L . Consequently, when ( λ, x ) ∈ C × C n is considered as an approximateeigenelement of P , it is natural to consider ( λ, Λ m − ⊗ x ) ∈ C × C mn as an approximateeigenelement of L and vice-versa. We denote the (unstructured) backward error of ( λ, Λ m − ⊗ x ) as an approximate eigenelement of L by η ( λ, Λ m − ⊗ x, L; v ) so as to show the dependenceof the backward error on the ansatz vector v. Similarly, we denote the structured backwardby η S M ( λ, Λ m − ⊗ x, L; v ) , where M ∈ { , F } . Now suppose that ( λ, x ) ∈ C × C n with x H x = 1 is an approximate eigenpair of P . Inview of (14) - (16), we only need to consider ansatz vectors v having unit norm. We use theinequality r m + 12 m ≤ k Λ m k k Λ m − k k ( λ, k ≤ Theorem 4.1
Let P ∈ P m ( C n × n ) be regular and L ∈ L (P) be a linearization of P corre-sponding to the normalized right ansatz vector v. Let ( λ, x ) ∈ C × C n be such that x H x = 1 . Set Λ m − := [ λ m − , . . . , λ, T . Then we have r m + 12 m ≤ η ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ . roof: By (1) and (14), we have η ( λ, Λ m − ⊗ x, L; v ) = k L( λ )(Λ m − ⊗ x ) k k (Λ m − ⊗ x ) k k ( λ, k = k v k k P( λ ) x k k (Λ m − ⊗ x ) k k ( λ, k = k v k k Λ m k k x k k (Λ m − ⊗ x ) k k ( λ, k η ( λ, x, P)= k Λ m k k Λ m − k k ( λ, k η ( λ, x, P) . Hence by (17) the desired result follows. (cid:4)
Theorem 4.1 shows that as far as the backward errors of approximate eigenelements ofP are concerned, any linearization from L (P) is as good as any other provided that thelinearization is associated to a normalized right ansatz vector. In contrast, restricting L in DL (P) (see, [20]), it is shown in [4] that the condition number of an eigenvalues λ of P isincreased at least by δ ( λ, v ) and at most by √ δ ( λ, v ) , where δ ( λ, v ) := k Λ m − k / | p ( x ; v ) | , see also [13].For a structured matrix polynomials, there exists infinitely many structured lineariza-tions, see [12, 21, 18]. For the structures we consider in this paper, we consider structuredlinearization from L (P) . For a ready reference, we summarize in Table 2 the condition onansatz vector for a structured linearization, see [12, 21]. The matrix Σ in Table 2 is given byΣ = diag { ( − m − , ( − m − , . . . , ( − } . S Structured Linearization ansatz vector sym sksymm v ∈ C m skew - sym skew-symm v ∈ C m T - even T -even Σ v = vT -odd Σ v = − vT - odd T -even Σ v = − vT - odd Σ v = v Herm
Herm v ∈ R m skew-Herm v ∈ i R m skew - Herm
Herm v ∈ i R m skew-Herm v ∈ R m H - even H -even Σ v = vT -odd Σ v = − vH - odd H -even Σ v = − vH -odd Σ v = v Table 2: Admissible ansatz vectors for structured linearizations.Recall that η ( λ, x, P) ≤ η S M ( λ, x, P) . Similarly, for a structured linearization from L (P)we have η ( λ, Λ m − ⊗ x, L; v ) ≤ η S M ( λ, Λ m − ⊗ x, L; v ) , where v is the ansatz vector. With aview to understanding the effect of structure preserving linearizations on the backward errorsof approximate eigenelements of structured matrix polynomials, in this section we compare η ( λ, x, P) and η S M ( λ, x, P) with η S M ( λ, x, L) . Corollary 4.2
Let P ∈ S and L ∈ L (P) be a structured linearization corresponding to thenormalized ansatz vector v. Then for M ∈ { , F } , we have η S M ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≥ r m + 12 m . roof: By Theorem 4.1 we have η S M ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≥ η ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≥ r m + 12 m . Hence the proof. (cid:4)
For a symmetric matrix polynomial P, any ansatz vector v yields a potential symmetriclinearization. Recall that an ansatz vector v is always assumed to be normalized, that is, k v k = 1 . We now show that structure preserving linearizations of symmetric and skew-symmetric matrix polynomials have almost no adverse effect on the backward errors of ap-proximate eigenelements.
Theorem 4.3
Let S be the space of symmetric matrix polynomials and P ∈ S . Let L ∈ L (P) be a symmetric linearization of P with normalized ansatz vector v. Finally, let ( λ, x ) ∈ C × C n be such that k x k = 1 . Then we have r m + 12 m ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η S F ( λ, x, P) ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ √ , r m + 12 m ≤ η S ( λ, Λ m − ⊗ x, L; v ) η S ( λ, x, P) = η ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ . Proof:
For the Frobenius norm, by Theorem 3.2 we have η S F ( λ, Λ m − ⊗ x, L; v ) η S F ( λ, x, P) = r k r k − | Λ Tm − v | k Λ m − k | x T r | p k r k − | x T r | · k Λ m k k Λ m − k k ( λ, k , where r := − P( λ ) x. Hence by (17) we have η S F ( λ, Λ m − ⊗ x, L; v ) η S F ( λ, x, P) ≥ q m +12 m . Next, since k r k ≤ r k r k − | Λ Tm − v | k Λ m − k | x T r | ≤ √ k r k , we have η S F ( λ, Λ m − ⊗ x, L; v ) η S F ( λ, x, P) ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ √ k Λ m k k Λ m − k k ( λ, k ≤ √ . Finally, by Theorem 3.2, we have structured and unstructured backward errors are thesame for the spectral norm. Hence the desired results follow from Theorem 4.1. (cid:4)
For skew-symmetric linearizations of skew-symmetric matrix polynomials, we have thefollowing result.
Theorem 4.4
Let S be the space of skew-symmetric matrix polynomials and P ∈ S . Let L ∈ L (P) be a skew-symmetric linearization of P with normalized ansatz vector v. Finally,let ( λ, x ) ∈ C × C n be such that k x k = 1 . Then for M ∈ { , F } we have r m + 12 m ≤ η S M ( λ, Λ m − ⊗ x, L; v ) η S M ( λ, x, P) = η ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ . Proof:
By Theorem 3.5 we have η S M ( λ, Λ m − ⊗ x, L; v ) η S M ( λ, x, P) = η ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) . (cid:4) Thus we conclude that for a symmetric/skew-symmetric matrix polynomial a structurepreserving linearization automatically ensures that the backward errors of approximate eigenele-ments are least affected by the conversion the polynomial eigenvalue problem into a generalizedeigenvalue problem of larger dimension. Moreover, as shown in [4] this choice also ensuresthat the linearization has a mild influence on the structured condition numbers of eigenvaluesof the polynomial. T -even and T -odd linearizations Now we analyze T -even and T -odd linearizations. Note that a T -even (resp., T -odd) polyno-mial admits T -even as well as T -odd linearizations which preserve the spectral symmetry ofthe T -even (resp., T -odd) polynomial. Theorem 4.5
Let P ∈ P m ( C n × n ) be a T -even polynomial and ( λ, x ) ∈ C × C n be such that k x k = 1 . Let S e ⊂ L (P) and S o ⊂ L (P) , respectively, denote the space of T -even and T -oddpencils. Finally, let L e ∈ S e (resp., L o ∈ S o ) be T -even (resp. T -odd) linearization of P with normalized ansatz vector v = Σ v (resp., v = − Σ v ). Then for M ∈ { , F } we have thefollowing.1. If | λ | ≤ then q m +12 m ≤ η S e M ( λ, Λ m − ⊗ x, L e ; v ) η ( λ, x, P) ≤ √ .
2. If | λ | ≥ then q m +12 m ≤ η S o M ( λ, Λ m − ⊗ x, L o ; v ) η ( λ, x, P) ≤ √ . Proof:
First consider the T -even linearization L e . Then by Theorem 3.7 we have η S e F ( λ, Λ m − ⊗ x, L e ; v ) η ( λ, x, P) = r k r k + ( | λ | − | Λ Tm − v | k Λ m − k | x T r | ! k Λ m k k r k k Λ m − k k ( λ, k , (18)where r := − P( λ ) x. Now for | λ | ≤ , we have k r k ≤ r k r k + ( | λ | − | Λ Tm − v | k Λ m − k | x T r | ≤ √ k r k . Henceby (17) we obtain the desired results for the Frobenius norm.Again by Theorem 3.7, we have η S e ( λ, Λ m − ⊗ x, L e ; v ) η ( λ, x, P) = r k r k + | λ | | Λ Tm − v | k Λ m − k | x T r | ! k Λ m k k r k k Λ m − k k ( λ, k . (19)Notice that k r k ≤ r k r k + | λ | | Λ Tm − v | k Λ m − k | x T r | ≤ p | λ | k r k for λ ∈ C . Hence by (17)we obtain the desired result for the spectral norm.Next, consider the T -odd linearization L o . Then by Theorem 3.10 we have η S o F ( λ, Λ m − ⊗ x, L o ; v ) η ( λ, x, P) = r k r k + ( | λ | − | Λ Tm − v | k Λ m − k | x T r | ! k Λ m k k r k k Λ m − k k (1 , λ ) k , (20)for λ = 0 . Now for | λ | ≥ , we have k r k ≤ s k r k + ( | λ | − − | Λ Tm − v | k Λ m − k | x T r | ≤ √ k r k . η S o ( λ, Λ m − ⊗ x, L o ; v ) η ( λ, x, P) = r k r k + | Λ Tm − v | | λ | k Λ m − k | x T r | ! k Λ m k k r k k Λ m − k k (1 , λ ) k (21)for λ = 0 . For | λ | ≥ , we have k r k ≤ r k r k + | λ | − | Λ Tm − v | k Λ m − k | x T r | ≤ √ k r k . Hence by (17)we obtain the desired result follows for the spectral norm. (cid:4)
Remark 4.6
We mention that the bounds in Theorem 4.5 also hold when P is T -odd withthe role of T -even and T -odd linearizations are reversed, that is, by interchanging the role of L e and L o we obtain the desired bounds. Next, comparing η S ( λ, x, P) with η S ( λ, x, L , v ) we have the following result. Theorem 4.7
Suppose that the assumptions of Theorem 4.5 hold. Let S ⊂ P m ( C n × n ) denotethe set of T -even polynomials. Then we have1. If | λ | ≤ q m +14 m ≤ η S e ( λ, Λ m − ⊗ x, L e ; v ) η S ( λ, x, P) ≤ √ .
2. If | λ | ≥ q m +14 m ≤ η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x, P) ≤ √ , when m is even and q m +12 m k (1 , λ ) k ≤ η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x, P) ≤ √ , when m is odd. Proof:
Note that the upper bounds follow from Theorem 4.5. We now derive the lowerbounds.First suppose that | λ | ≤ . Then it is easy to see that k ( I − Π e )(Λ m ) k ≤ k Π e (Λ m ) k . Hence by Theorem 3.7 we have η S ( λ, x, P) ≤ k r k k Λ m k s k ( I − Π e )(Λ m ) k k Π e (Λ m ) k ≤ √ k r k k Λ m k . On the other hand, by (19) we have η S e ( λ, Λ m − ⊗ x, L e ; v ) ≥ k r k k Λ m − k k (1 , λ ) k . Conse-quently, by (17) we have η S e ( λ, Λ m − ⊗ x, L e ; v ) η S ( λ, x, P) ≥ k Λ m k √ k Λ m − k k (1 .λ ) k ≥ r m + 1 m . Next suppose that | λ | ≥ T -odd linearization L o . Then it is easy tocheck that k ( I − Π e )(Λ m ) k ≤ k Π e (Λ m ) k when m is even and the desired result followsby similar arguments as above. Now suppose that m is odd. Then it is easy to see that k ( I − Π e )(Λ m ) k = | λ | k Π e (Λ m ) k . Hence by Theorem 3.7 we have η S ( λ, x, P) ≤ k r k k Λ m k s k ( I − Π e )(Λ m ) k k Π e (Λ m ) k ≤ p | λ | k r k k Λ m k . Further by (21) we have η S o ( λ, Λ m − ⊗ x, L o ; v ) ≥ k r k k Λ m − k k (1 , λ ) k . Hence by (17) we have η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x, P) ≥ k Λ m k k Λ m − k k (1 .λ ) k ≥ √ k (1 , λ ) k r m + 1 m . This completes the proof. (cid:4)
For T -odd polynomials, we have the following result.20 heorem 4.8 Let S ⊂ P m ( C n × n ) denote the space of T -odd matrix polynomials and P ∈ S . Let S e ⊂ L (P) and S o ⊂ L (P) , respectively, denote the space of T -even and T -odd pencils.Finally, let L e ∈ S e (resp., L o ∈ S o ) be T -even (resp. T -odd) linearization of P with normalizedansatz vector v = Σ v (resp., v = − Σ v ). Then for ( λ, x ) ∈ C × C n with k x k = 1 , we have thefollowing.1. If | λ | ≤ q m +16 m ≤ η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x, P) ≤ .
2. If | λ | ≥ k ( √ , λ ) k q m +1 m ≤ η S e ( λ, Λ m − ⊗ x, L e ; v ) η S ( λ, x, P) ≤ , when m is even and q m +14 m ≤ η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x, P) ≤ , when m is odd. Proof:
By Theorem 3.10 we have η S ( λ, x ; P) = 1 k Λ m k s k r k + k Π e (Λ m ) k k ( I − Π e )(Λ m ) k | x T r | . Itis easy to see that | λ | | λ | ≥ k ( I − Π e )(Λ m ) k k Λ m k (22)with equality holds for odd m. Now, by (21) we have η S o ( λ, Λ m − ⊗ x, L o ; v ) = 1 k Λ m − k k (1 , λ ) k s k r k + | λ | − | Λ Tm − v | k Λ m − k | x T r | . Since by (22), k Π e (Λ m ) k k ( I − Π e )(Λ m ) k ≥ | λ | − with equality holds for odd m, we have η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x ; P) ≤ . (23)Also it is easy to check that k ( I − Π e )(Λ m ) k ≤ k Λ m k ≤ √ k Π e (Λ m ) k whenever | λ | ≤ . Consequently we have k ( I − Π e )(Λ m ) k k Π e (Λ m ) k ≤ √ . This yields η S o ( λ, Λ m − ⊗ x, L o ; v ) η S ( λ, x ; P) ≥ r m + 16 m . Next suppose that | λ | ≥ . If m is even then its obvious that k Π e (Λ m ) k k ( I − Π e )(Λ m ) k ≥ | λ | . Henceby (19) and (17) we have η S e ( λ, Λ m − ⊗ x, L e ; v ) η S ( λ, x ; P) ≤ k Λ m k k Λ m − k k (1 , λ ) k r k r k + | λ | | Λ Tm − v | k Λ m − k | x T r | p k r k + | λ | | x T r | ≤ . Further using the fact k Π e (Λ m ) k k ( I − Π e )(Λ m ) k ≤ | λ | we have η S e ( λ, Λ m − ⊗ x, L e ; v ) η S ( λ, x ; P) ≥ k ( √ , λ ) k r m + 1 m . On the other hand, if m is odd and | λ | ≥ k Π e (Λ m ) k k ( I − Π e )(Λ m ) k = | λ | ≤ . This completes the proof. (cid:4) T -even matrix polynomialP , it is advisable to solve T -even as well as T -odd linearizations of P and then choose acomputed eigenpair ( λ, x ) from T -even or T -odd linearization according as | λ | ≤ | λ | ≥ . In contrast, when P is T -odd it is advisable to choose ( λ, x ) from T -even linearization onlywhen | λ | ≥ λ, x ) from T -odd linearization of P . This choice ensures that the linearizations have almost no adverse effect on the backward errorof the computed eigenelement ( λ, x ) . We arrived at the same conclusion in [4] by analyzing theeffect of structure preserving linearizations on the structured condition numbers of eigenvaluesof the polynomial P . H -even linearizations First, we consider Hermitian matrix polynomials. Note that a Hermitian matrix polynomialadmits Hermitian and skew-Hermitian linearizations both preserving the spectral symmetryof the Hermitian polynomial.
Theorem 4.9
Let P ∈ P m ( C n × n ) be Hermitian and ( λ, x ) ∈ C × C n be such that k x k = 1 . Let S ∈ { Herm , skew - Herm } and L ∈ S be a linearization of P with normalized ansatz vector v. If λ ∈ R then we have r m + 12 m ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η Herm F ( λ, x, P) ≤ √ and r m + 12 m ≤ η S ( λ, Λ m − ⊗ x, L; v ) η Herm ( λ, x, P) ≤ . The same bounds hold when P is skew-Hermitian. Proof:
First, suppose that S = Herm so that L is a Hermitian linearization of P . For λ ∈ R , by Theorem 3.12, we have η S F ( λ, Λ m − ⊗ x, L; v ) η Herm F ( λ, x, P) = r k r k − | Λ Hm − v | k Λ m − k | x H r | p k r k − | x H r | · k Λ m k k Λ m − k k ( λ, k , where r := − P( λ ) x. Hence by (17) we have η S F ( λ, Λ m − ⊗ x, L; v ) η Herm F ( λ, x, P) ≥ q m +12 m . Next, since k r k ≤ r k r k − | Λ Hm − v | k Λ m − k | x H r | ≤ √ k r k , we have η S F ( λ, Λ m − ⊗ x, L; v ) η Herm F ( λ, x, P) ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ √ k Λ m k k Λ m − k k ( λ, k ≤ √ . For the spectral norm, by Theorem 3.12, structured and unstructured backward errors arethe same when λ ∈ R . Hence the desired results follow from Theorem 4.1.Finally, since the backward errors are the same for Hermitian and skew-Hermitian pencils,the above hounds obviously hold for the case when S = skew - Herm . (cid:4) This shows that a structured linearization of a Hermitian matrix polynomial does not haveadverse effect on the backward errors of approximate eigenelements when the approximateeigenvalues are real. On the other hand, when the approximate eigenvalues are complex,the structured backward errors are not amenable to easy comparisons. Indeed, under theassumptions of Theorem 4.9, when λ ∈ C \ R by Theorem 3.12 a little calculation shows that η S F ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ s k b r k k r k and η S ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ s k b r k k r k , r := − P( λ ) x and b r = r h := (cid:20) re λ im λ (cid:21) † (cid:20) re (Λ Hm − vx H P( λ ) x ) im (Λ Hm − vx H P( λ ) x ) (cid:21) when S = Herm , and b r = r s := (cid:20) − im λ re λ (cid:21) † (cid:20) re (Λ Hm − vx H P( λ ) x ) im (Λ Hm − vx H P( λ ) x ) (cid:21) when S = skew - Herm . Next we consider linearizations of H -even polynomials. Note that an H -even polynomialadmits H -even as well as H -odd linearizations and both have the same spectral symmetry asthat of the polynomial. For purely imaginary eigenvalues of an H -even or H -odd polynomial,we have the following result. Theorem 4.10
Let P ∈ P m ( C n × n ) be H -even and ( λ, x ) ∈ C × C n be such that k x k = 1 . Let S ∈ { H - even , H - odd } and L ∈ S be a linearization of P with normalized ansatz vector v. If λ ∈ i R then we have r m + 12 m ≤ η S F ( λ, Λ m − ⊗ x, L; v ) η H - even F ( λ, x, P) ≤ √ and r m + 12 m ≤ η S ( λ, Λ m − ⊗ x, L; v ) η H - even ( λ, x, P) ≤ . The same bounds hold when P is H -odd. Proof:
The proof is exactly the same as that of Theorem 4.9 and follows from Theorem 3.14. (cid:4)
This shows that a structured linearization of an H -even polynomial has least influenceon the backward errors of approximate eigenelements when the approximate eigenvalues arepurely imaginary. On the other hand, when the approximate eigenvalues are not purelyimaginary, the structured backward errors of approximate eigenelements are not amenableto easy comparisons. Indeed, under the assumptions of Theorem 4.10, when λ ∈ C \ i R byTheorem 3.14, we have η S F ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ s k b r k k r k and η S ( λ, Λ m − ⊗ x, L; v ) η ( λ, x, P) ≤ s k b r k k r k , where r := − P( λ ) x and b r = r s when S = H - even , and b r = r h when S = H - odd . The obvious conclusion that we can draw is that real eigenvalues of a Hermitian/skew-Hermitian matrix polynomial can be computed either by solving a Hermitian or a skew-Hermitian linearization. However, for non real eigenvalues it may be a good idea to solveHermitian as well as skew-Hermitian linearizations and choose an eigenpair ( λ, x ) from Her-mitian or skew-Hermitian linearization according as r h ≤ r s or r s ≤ r h . Similar conclusionholds for H -even/ H -odd matrix polynomials. These observations are consistent with thosemade in [4] by analyzing the structured condition numbers of eigenvalues. Let P be a regular polynomial. For λ ∈ C , the backward error of λ as an approximate eigen-value of P is given by η ( λ, P) := min { η ( λ, x, P) : x ∈ C n and k x k = 1 } . Since η ( λ, x, P) = k P( λ ) x k / k x k k Λ m k , it follows that for the spectral as well as for the Frobenius norms on C n × n , we have η ( λ, P) := σ min (P( λ )) k Λ m k . Similarly, for M ∈ { , F } we define structured backward error of an approximate eigenvalue λ of P ∈ S by η S M ( λ, P) := min { η S M ( λ, x, P) : x ∈ C n and k x k = 1 } . In this section, we make an attempt to determine η S M ( λ, P) . The backward errors of approx-imate eigenvalues and pseudospectra of a polynomial are closely related. For ǫ > , the23nstructured ǫ -pseudospectrum of P, denoted by σ ǫ (P) , is given by (see [5, 6]) σ ǫ (P) = [ |||△ P ||| M ≤ ǫ { σ (P + △ P) : △ P ∈ P m ( C n × n ) } . Obviously, we have σ ǫ (P) = { z ∈ C : η ( z, P) ≤ ǫ } , assuming, for simplicity, that ∞ / ∈ σ ǫ (P) , see [5, 6]. For the sake of simplicity in this section we make an implicit assumption that ∞ / ∈ σ ǫ (P) . Observe that since η ( λ, P) is the same for the spectral norm and the Frobeniusnorm, we conclude that σ ǫ (P) is the same for the spectral and the Frobenius norms. Similarly,when P ∈ S , we define the structured ǫ -pseudospectrum of P , denoted by σ S ǫ (P) , by σ S ǫ (P) := [ |||△ P ||| M ≤ ǫ { σ (P + △ P) : △ P ∈ S } . Then it follows that σ S ǫ (P) = { z ∈ C : η S M ( λ, P) ≤ ǫ } . Theorem 5.1
Let S ∈ { sym , skew - sym } and P ∈ S . Then for the spectral norm, we have η S ( λ, P) = η ( λ, P) and σ S ǫ (P) = σ ǫ (P) . On the other hand, for the Frobenius norm, wehave η S F ( λ, P) = √ η ( λ, P) and σ S ǫ (P) = σ ǫ/ √ (P) when S = skew - sym , and η S F ( λ, P) = η ( λ, P) and σ S ǫ (P) = σ ǫ (P) when S = sym . Proof:
For the spectral norm, by Theorem 3.2, we have η S ( λ, x, P) = η ( λ, x, P) for all x. Consequently, we have η S ( λ, P) = η ( λ, P) . Hence the result follows.For the Frobenius norm, the result follows from Theorem 3.5 when P is skew-symmetric.So, suppose that P is symmetric. Then P( λ ) ∈ C n × n is symmetric. Consider the Takagifactorization P ( λ ) = U Σ U T , where U is unitary and Σ is a diagonal matrix containingsingular values of P( λ ) (appear in descending order). Set σ := Σ( n, n ) and u := U (: , n ) . Thenwe have P( λ ) u = σu. Now define △ A j := − λ j σ uu T k Λ m k , and consider the polynomial △ P( z ) = P mj =0 z j △ A j . Then △ P is symmetric and P( λ ) u + △ P( λ ) u = 0 . Note that for M ∈ { , F } we have η S M ( λ, P) ≤ |||△ P ||| M = σ k Λ m k = η ( λ, P) and hence σ ǫ (P) = σ S ǫ (P) . This completes the proof. (cid:4)
When P is symmetric, the above proof shows how to construct a symmetric polynomial △ P such that λ ∈ σ (P + △ P) and |||△ P ||| M = η S M ( λ, P) . When P is skew-symmetric, usingTakagi factorization of the complex skew-symmetric matrix P( λ ) , one can construct a skew-symmetric polynomial △ P such that λ ∈ σ (P+ △ P) and |||△ P ||| M = η S M ( λ, P) . Indeed, considerthe Takagi factorization P( λ ) = U diag( d , · · · , d m ) U T , where U is unitary, d j := (cid:20) s j − s j (cid:21) , s j ∈ C is nonzero and | s j | are singular values of P( λ ) . Here the blocks d j appear in descending order of magnitude of | s j | . Note that P( λ ) U = U diag( d , · · · , d m ) . Let u := U (: , n − n ) . Then P( λ ) u = ud m = ud m u T u. Now define △ A j := − λ j ud m u T k v λ k and consider the pencil △ P( z ) = P mj =0 z j △ A j . Then △ P is skew-symmetric and P( λ ) u + △ P( λ ) u = 0 . Further, we have η S ( λ, P) = |||△ P ||| = σ min (P( λ )) k Λ m k = η ( λ, P) , η S F ( λ, P) = |||△ P ||| F = √ σ min (P( λ )) k Λ m k = √ η ( λ, P) .
24e denote the unit circle in C by T , that T := { z ∈ C : | z | = 1 } . Then for the T -even or T -odd polynomials we have the following result. Theorem 5.2
Let S ∈ { T -even, T -odd } and P ∈ S and m is odd. Then for λ ∈ T and theFrobenius norm we have η S F ( λ, P) = √ η ( λ, P) and σ S ǫ (P) ∩ T = σ ǫ/ √ (P) ∩ T . Proof:
Let λ ∈ T . Then by Theorem 3.7 and Theorem 3.10, η S F ( λ, x, P) = √ k P( λ ) x k / k Λ m k for all x such that k x k = 1 . Hence taking minimum over k x k = 1 , we obtain the desiredresults. (cid:4) Theorem 5.3
Let S ∈ { Herm , skew - Herm } and P ∈ S . Then for M ∈ { , F } and λ ∈ R , wehave η S M ( λ, P) = η ( λ, P) and σ S ǫ (P) ∩ R = σ ǫ (P) ∩ R . Proof:
Note that P( λ ) is either Hermitian or skew-Hermitian. Let ( µ, u ) be an eigenpair ofthe matrix P( λ ) such that | µ | = σ min (P( λ )) and u H u = 1 . Then P( λ ) u = µu. Define △ A j := − λ j µ uu H k Λ m k and consider the polynomial △ P( z ) = P mj =0 z j △ A j . Then △ P ∈ S and λ ∈ σ (P + △ P) . Further, we have |||△ P ||| M = σ min (P( λ )) k Λ m k . Hence the result follows. (cid:4)
Theorem 5.4
Let S ∈ { H -even, H -odd } and P ∈ S . Then for M ∈ { , F } and λ ∈ i R , wehave η S M ( λ, P) = η ( λ, P) and σ S ǫ (P) ∩ i R = σ ǫ (P) ∩ i R . Proof:
Note for λ ∈ i R , then the matrix P( λ ) is again either is Hermitian or skew-Hermitian.Hence the result follows from the proof of Theorem 5.3. (cid:4) We mention that the above results can be easily extended to the case of general structuredpolynomials where the coefficients matrices are elements of Jordan and/or Lie algebras.Finally, we mention that the partial equality σ S ǫ (L) ∩ Ω = σ ǫ (L) ∩ Ω , for an appropriateΩ ⊂ C , and the minimal perturbations constructed above as well as in section 3 are expectedto be key tools for solution of certain structured distance problems, see [7]. For illustration,consider an H -even polynomial P . We have seen that the eigenvalues of P have Hamiltonianspectral symmetry, that is, the spectrum of P is symmetric with respect to the imaginary axisin the complex plane and thus appear in the pair ( λ, − λ ) . Obviously the spectral symmetrydegenerates if there are purely imaginary eigenvalues. Often in practice the polynomial Pis obtained by an approximation of the exact problem. Thus it may be the case that eventhough the exact problem has no purely imaginary eigenvalues but due to approximation errorthe polynomial P may have a few purely imaginary eigenvalues. So, the task is to constructa minimal perturbation △ P such that P + △ P is H -even and that the perturbed polynomialP + △ P has no eigenvalues on the imaginary axis. On the other hand, it may also be thecase that all eigenvalues of the exact problem are purely imaginary (e.g. gyroscopic system)but due to approximation error the polynomial P has a few eigenvalues off the imaginaryaxis. In such a case the task is to construct a minimal perturbation △ P such that P + △ P is H -even and that all eigenvalues of P + △ P are purely imaginary. The minimal perturbationsso constructed and the partial equality of between structured and unstructured pseudospectraso established are expected to be key tools in solving these problems and will be investigatedelsewhere.
Conclusion.
We have derived backward errors of approximate eigenelements of structuredmatrix polynomials. We have constructed minimal structured perturbations that induce theapproximate eigenelements as exact eigenelements of the perturbed polynomials. The minimalperturbations so constructed are expected to be useful in analyzing the evolution of eigenval-ues of structured polynomials under structure preserving perturbations. We have analyzed25he influence of structure preserving linearizations on the approximate eigenelements of struc-tured matrix polynomials. Also, we have provided a recipe for selecting structure preservinglinearizations so that the linearizations have almost no adverse effect on the approximateeigenelements of the structured matrix polynomials. We have briefly analyzed structuredpseudospectra of structured matrix polynomials and have shown that partial equality be-tween structured and unstructured pseudospectra holds for certain structured polynomials.These results are expected to be useful for constructing minimal structured perturbationsthat move eigenvalues of the structured polynomials along certain directions which in turnare expected to be key tools for solving certain structured distance problems.
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