Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices
BBlock Tridiagonal Reduction of Perturbed Normal and RankStructured Matrices ∗ Roberto Bevilacqua, Gianna M. Del Corso and Luca Gemignani † May 2, 2013
Abstract
It is well known that if a matrix A ∈ C n × n solves the matrix equation f ( A , A H ) =
0, where f ( x , y ) is alinear bivariate polynomial, then A is normal; A and A H can be simultaneously reduced in a finite numberof operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located ona straight line in the complex plane. In this paper we present some generalizations of these properties foralmost normal matrices which satisfy certain quadratic matrix equations arising in the study of structuredeigenvalue problems for perturbed Hermitian and unitary matrices. Keywords
Block tridiagonal reduction, rank structured matrix, block Lanczos algorithm
MSC
Normal matrices play an important theoretical role in the field of numerical linear algebra. A squarecomplex matrix is called normal if A H A − AA H = , where A H is the conjugate transpose of A . Polyanalytic polynomials are functions of the form p ( z ) = ∑ kj = h k − j ( z ) ¯ z j , where h j ( z ) , 0 ≤ j ≤ k , are complex polynomials of degree less than or equal to j . A poly-analytic polynomial of minimal total degree that annihilates A , i.e., such that p ( A ) =
0, is called a minimalpolyanalytic polynomial of A [15]. Over the years many equivalent conditions have been found [12, 7], andit has been discovered that the class of normal matrices can be partitioned in accordance with a parameter s ∈ N , s ≤ n −
1, where s is the minimal degree of a particular polyanalytic polynomial p s ( z ) = ¯ z − n s ( z ) such that p s ( A ) = A H − n s ( A ) =
0, and n s ( z ) is a polynomial of degree s .For a normal matrix the assumption of being banded imposes strong constraints on the localization ofthe spectrum and the degree of minimal polyanalytic polynomials. It is well known that the minimal poly-analytic polynomial of an irreducible normal tridiagonal matrix has degree one and, moreover, the spectrumof the matrix is located on a straight line in the complex plane [9, 14]. Generalizations of these propertiesto normal matrices with symmetric band structure are provided in [16]. Nonsymmetric structures are con-sidered in the papers [17, 8] where it is shown that the customary Hessenberg reduction procedure appliedto a normal matrix always returns a banded matrix with upper bandwidth at most k if and only if s ≤ k . Away to arrive at the Hessenberg form is using the Arnoldi method which amounts to construct a sequenceof nested Krylov subspaces. A symmetric variation of the Arnoldi method named generalized Lanczosprocedure is devised in [6] and applied in [6, 15] and [11] for the block tridiagonal reduction of normaland perturbed normal matrices, respectively. The reduction is rational –up to square root calculations–and finite but not computationally appealing since it essentially reduces to the orthonormalization of thesequence of generalized powers A j A kH v , j + k = m , m ≥ ∗ This work was partially supported by GNCS-INDAM, grant ”Equazioni e funzioni di Matrici” † Universit`a di Pisa, Dipartimento di Informatica, Largo Pontecorvo, 3, 56127 Pisa, Italy, email: { bevilacq, delcorso,l.gemignani } @di.unipi.it a r X i v : . [ m a t h . NA ] J un n [1] the class of almost normal matrices is introduced, that is the class of matrices for which [ A , A H ] = A H A − AA H = CA − AC for a low rank matrix C . In the framework of operator theory conditions upon thecommutator [ A , A H ] are widely used in the study of structural properties of hypernormal operators [18].Our interest in the class of almost normal matrices stems from the analysis of fast eigenvalue algorithmsfor rank–structured matrices. If A is a rank–one correction of a Hermitian or unitary matrix than A satisfies [ A , A H ] = CA − AC for a matrix C of rank at most 2. Furthermore, this matrix C is involved in the descriptionof the rank structure of the matrices generated starting from A under the QR process [2, 4, 19]. Thus theproblem of simultaneously reducing both A and C to symmetric band structure is theoretically interestingbut it also might be beneficial for the design of fast effective eigenvalue algorithms for these matrices.Furthermore, condensed representations expressed in terms of block matrices [5] or the product of simplermatrices [3, 20] tends to become inefficient as the length of the perturbation increases [10]. The exploitationof condensed representations in banded form can circumvent these difficulties.In [1] it is shown that we can always find an almost normal block tridiagonal matrix with blocks ofsize 2 which fulfills the commutator equation for a certain C with rank ( C ) =
1. Although an algorithmicconstruction of a block tridiagonal solution is given, no efficient computational method is described in thatpaper for the block tridiagonal reduction of a prescribed solution of the equation. In this contribution, wefirst propose an algorithm based on the application of the block Lanczos method to the matrix A + A H starting from a suitable set of vectors associated with the range of the commutator [ A , A H ] for the blocktridiagonal reduction of an eligible solution of the commutator equation. Then we generalize the approachto the case where rank ( C ) = A is a rank–one modification of particular normal matrices whose eigenvalues lie ona real algebraic curve of degree 2. In the latter case the matrix C of rank at most 2 could not exist and,therefore, the analysis of this configuration is useful to put in evidence the consequences of such a missing. In this section we discuss the reduction to block tridiagonal form of almost normal matrices.
Definition 1.
Let A be an n × n matrix. If there exists a rank-k matrix C such that [ A , A H ] = A H A − AA H = CA − AC , we say that A is a k-almost normal matrix. Denote by ∆ ( A ) : = [ A , A H ] = A H A − AA H the commutator of A and by S the range of ∆ ( A ) . It is clearthat if, for a given C , any solution of the nonlinear matrix equation [ X , X H ] = X H X − XX H = CX − XC , C , X ∈ C n , (1)exists, then it is not unique. Indeed, if A is an almost-normal matrix such that ∆ ( A ) = CA − AC then B = A + γ I , with a complex constant γ , is almost normal as well and ∆ ( B ) = ∆ ( A ) = CB − BC . In [1]the structure of almost normal matrices with rank–one perturbation is studied by showing that a block-tridiagonal matrix with 2 × C of rank one. Here wetake a different look at the problem by asking whether a solution of (1) for a given C can be reduced toblock tridiagonal form.The block Lanczos algorithm is a technique for reducing a Hermitian matrix H ∈ C N × n to block tridi-agonal form. There are many variants of the basic block Lanczos procedure. The method stated below isin the spirit of the block Lanczos algorithm described in [13].2 rocedure Block LanczosInput : H , Z ∈ C n × (cid:96) nonzero, (cid:96) ≤ n ; [ Q , Σ , V ] = svd ( Z ) ; s = rank ( Σ ) ; U ( : , s ) = Q ( : , s ) ; s = , s = s ; while s < nW = A H · U ( : , s ) ; T ( s : s , s : s ) = ( U ( : , s )) H · W ; if s = W = W − U ( : , s : s ) · T ( s : s , s : s ) ; else W = W − U ( : , s : s ) · T ( s : s , s : s ) ; W = W − U ( : , ˆ s : ˆ s ) · T ( ˆ s : ˆ s , s : s ) ; end [ Q , Σ , V ] = svd ( W ) ; s new = rank ( Σ ) ; if s new = disp (’premature stop’); return ; else Σ = Σ ( s new , s new ) · ( V ( : , s )) H ;ˆ s = s , ˆ s = s , s = s + , s = s + s new ; U ( : , s : s ) = Q ( : , s new ) , T ( s : s , ˆ s : ˆ s ) = Σ ( s new , s ) ; T ( ˆ s : ˆ s , s : s ) = ( T ( s : s , ˆ s : ˆ s )) H , s = s new ; endend T ( s : s , s : s ) = ( U ( : , s : s )) H · A H · U ( : , s : s ) ; The procedure, when terminates without a premature stop, produces a block-tridiagonal matrix and aunitary matrix U such that U H H U = T = A B H B A . . .. . . . . . B Hp − B p − A p , where A k ∈ C i k × i k , B k ∈ C i k + × i k , and (cid:96) ≥ i k ≥ i k + , i + i + · · · + i p = n . In fact, the size of the blocks canpossibly shrink when the rank of the matrices W is less than (cid:96) .Let Z ∈ C n × (cid:96) , and denote by K j ( H , Z ) the block Krylov subspace generated by the column vectorsin Z , that is the space spanned by the columns of the matrices Z , HZ , H Z , . . . , H j − Z . It is well knownthat the classical Lanczos process builds an orthonormal basis for the Krylov subspace K n − ( H , z ) , for z = α U ( : , ) . Similarly, when the block Lanczos process does not break down, span { U ( : , ) , U ( : , ) . . . , U ( : , n ) } = K j ( H , Z ) for j such that dim ( K j ( H , Z )) = n . When the block-Lanczos procedure terminates beforecompletion it means that K j ( H , Z ) is an invariant subspace and a null matrix W has been found in the aboveprocedure. In this case the procedure has to be restarted and the final matrix T is block diagonal. As anexample, we can consider the matrix H = U + U H , where U is the Fourier matrix U : = F n = √ n Ω n oforder n = m . Due to the relation Ω n = n Π , where Π is a suitable symmetric permutation matrix, it is foundthat for any starting vector z the Block Lanczos procedure applied with Z = [ z | U z ] breaks down within thefirst three steps. In this case the reduction scheme has to be restarted and the initial matrix can be convertedinto the direct sum of diagonal blocks. In this section we consider the case where the matrix A ∈ C n × n solves (1) for a prescribed nonzero matrix C of rank one, that is C = uv H , u , v (cid:54) = . We show that A can be unitarily converted to a block tridiagonalform with blocks of size at most 2 by applying the block-Lanczos procedure starting from a basis of S thecolumn space of ∆ ( A ) . 3et us introduce the Hermitian and antihermitian part of A denoted as A H : = A + A H , A AH : = A − A H . Observe that ∆ ( A ) = A H A − AA H = ( A H A AH − A AH A H ) . In the next theorems we prove that the Krylov subspace of A H obtained starting from a basis of S , thecolumn space of ∆ ( A ) , coincides with the Krylov space of A AH , and hence with that of A . We first needsome technical lemmas. Lemma 1.
Let A be a 1-almost normal matrix, and let C = uv H be a rank–one matrix such that ∆ ( A ) = CA − AC is nonzero. Then ∆ ( A ) has rank two and ( u , A u ) and ( v , A H v ) are two bases of S . Moreover, if u and v are linearly independent then ( u , v ) is a basis for S as well.Proof. Note that CA − AC = uv H A − A uv H , ∆ ( A ) is Hermitian and, therefore, ∆ ( A ) has rank two and S = span { u , A u } . Because of the symmetry of ∆ ( A ) , ( v , A H v ) is a basis of S as well. Moreover, if u and v are linearly independent they form a basis for S since both vectors belong to S . Lemma 2.
Let A be a 1-almost normal matrix with C = uv H , where u and v are linearly independent.Then we have A A kH u = k ∑ j = λ ( k ) j A jH u + k ∑ j = µ ( k ) j A jH v , k = , , . . . ; and similarly A H A kH v = k ∑ j = ˆ λ ( k ) j A jH u + k ∑ j = ˆ µ ( k ) j A jH v , k = , , . . . . Proof.
We prove the first case by induction on k . Observe that it holds A H A − AA H = ∆ ( A ) , which gives AA H u = A H A u − ∆ ( A ) u . From S = span { u , v } , ∆ ( A ) u ∈ S and A u ∈ S we deduce the relation for k =
1. Then assume the thesis istrue for k and let prove it for k +
1. Denote by x = A kH u , we have A A k + H u = A A H x = A H A x − ∆ ( A ) x . Since ∆ ( A ) x ∈ S , applying the inductive hypothesis we get the thesis. The proof of the second case proceedsanalogously by using A H A H − A H A H = ∆ ( A ) . Lemma 3.
Let Z = [ z | . . . | z (cid:96) ] ∈ C n × (cid:96) and X ∈ C n × (cid:96) such that span { Z } : = span { z , . . . , z (cid:96) } = span { X } ,then K j ( A , Z ) = K j ( A , X ) .Proof. If span { Z } = span { X } , then there exists a square nonsingular matrix B such that Z = X B . Let u ∈ K j ( A , Z ) , then we can write u as a linear combination of the vectors of K j ( A , Z ) , that is there exists a ( j + ) s vector a such that u = [ Z , AZ , . . . , A j Z ] a = [ XB , AXB , . . . , A j XB ] a == [ X , AX , . . . , A j X ] B B . . . B a = [ X , AX , . . . , A j X ] b ∈ K j ( A , X ) , b = ( I j + ⊗ B ) a . We denote as K j ( A , < z , . . . , z l > ) the Krylov subspace of A generated starting from any initial matrix X ∈ C n × (cid:96) satisfying span { z , . . . , z (cid:96) } = span { X } .The main result of this section is the following. Theorem 4.
Let A be a 1-almost normal matrix with C = uv H . If u and v are linearly independent then K j ( A AH , < u , v > ) ⊆ K j ( A H , < u , v > ) for each j. Thus, if the block Lanczos process does not breakdown prematurely, U H A H U e U H A AH U are block-tridiagonal and, hence, U H AU is block tridiagonal aswell. The size of the blocks is at most two.Proof.
The proof is by induction on j . For j =
1, let x ∈ K ( A AH , < u , v > ) , we need to prove that x ∈ K ( A H , < u , v > ) . Since x ∈ K ( A AH , < u , v > ) , then x ∈ span { u , v , A AH u , A AH v ) . It is enough toprove that A AH u ∈ K ( A H , < u , v > ) and A AH v ∈ K ( A H , < u , v > ) . From A AH + A H = A , A AH − A H = − A H (2)we obtain that A AH u = − A H u + A u . Since from Lemma 1 A u ⊆ span { u , v } ∈ K ( A H , < u , v > ) , we conclude that A AH u ∈ K ( A H , < u , v > ) . Similarly we find that A AH v = A H v − A H v ∈ K ( A H , < u , v > ) . Assume now that the thesis holds for j and prove it for j +
1. For the linearity of the Krylov subspaceswe can prove the thesis on the monomials and for each of the starting vectors u and v . Let x = A jAH u . Wehave A j + AH u = A AH x Since by inductive hypothesis x ∈ K j ( A H , < u , v > ) , x = ∑ jk = α k A kH u + ∑ jk = β k A kH v , and using (2) weobtain that A j + AH u = A AH x = ( − A H + A ) j ∑ k = α k A kH u + ( A H − A H ) j ∑ k = β k A kH v = − j + ∑ k = α k A kH u + j + ∑ k = β k A kH v + j ∑ k = α k AA kH u − j ∑ k = β k A H A kH v . By applying Lemma 2 to each term of the form AA kH u and A H A kH v in the previous relation we obtain that A j + AH u ∈ K j + ( A H , , u , v > ) . With a similar technique we prove that A j + AH v ∈ K j + ( A H , < u , v > ) .Lemma 3, provided u and v are linearly independent, proves that we can apply the block Lanczosprocedure to any pair of linearly independent vectors in S . The remaining case where u and v are notlinearly independent, that is the rank–one correction has the form C = α uu H , can treated as follows. Theorem 5.
Let A be a 1-almost normal matrix with C = α uu H . Then K j ( A AH , u ) ⊆ K j ( A H , u ) . Hence,if breakdown does not occur, the classic Lanczos process applied to A H with starting vector u returns aunitary matrix U which reduces A AH to tridiagonal form and therefore also U H AU is tridiagonal.Proof.
Set B = − i ¯ α A obtaining for B the following relation B H B − BB H = ˆ CB − B ˆ C , with ˆ C = i uu H , that is with an antihermitian correction. Since ∆ ( B ) is hermitian, we obtainˆ CB − B ˆ C = B H ˆ C H − ˆ C H B H = − B H ˆ C + ˆ CB H , and hence ˆ CB AH = B AH ˆ C , meaning that span { B AH u } ⊆ span { u } . This proves that K j ( B AH , u ) ⊆ span { u } ⊆ K j ( B H , u ) , and hencethat B AH is brought in tridiagonal form by means of the same unitary matrix which tridiagonalizes B H .Then B and A are brought to tridiagonal form by the same U .5ote that Theorem 5 states that any 1-almost normal matrix with C = α uu H can be unitarily transformedinto tridiagonal form. The class of 2-almost normal matrices is a richer and more interesting class. For example, rank–oneperturbations of unitary matrices, such as the companion matrix, belong to this class. Also generalizedcompanion matrices for polynomials expressed in the Chebyshev basis [4, 19] can be viewed as rank–oneperturbation of Hermitian matrices and are 2-almost normal.Assume A H A − AA H = CA − AC , where C = uv H + xy H . Note that dim ( S ) ≤
4. If the column spaceof ∆ ( A ) has dimension exactly 4 then possible bases for S are < u , x , A u , A x > , < v , y , A H v , A H y > and < u , v , x , y > when the four vectors are linearly independent.A theorem analogous to Theorem 4 for 2-almost normal matrices uses a generalization of Lemmas 1and 2. Lemma 6.
Let A be a 2-almost normal matrix, and let C = UV H , with U , V ∈ C n × be a rank- matrixsuch that ∆ ( A ) = CA − AC has rank . Then, the columns of the matrices [ U , AU ] and of [ V , A H V ] span thespace S . Moreover, if rank ([ U , V ]) = the columns of the matrix [ U , V ] form a basis for S as well. Similarly Lemma 2 can be generalized replacing the vectors u and v with two n × Lemma 7.
Let A be a 2-almost normal matrix with C = UV H , with U , V ∈ C n × , with rank ( ∆ ( A )) = andrank ([ U , V ]) = . Then we have A A jH U ∈ K j ( A H , [ U , V ]) j = , , . . . and similarly A H A jH V ∈ K j ( A H , [ U , V ]) j = , , . . . . We are now ready for the desired generalization of the main result. The proof is similar to that ofTheorem 4 and it is omitted here.
Theorem 8.
Let A be a 2-almost normal matrix with C = UV H , with U , V ∈ C n × . If rank ([ U , V ]) = andrank ( ∆ ( A )) = , then we have K j ( A AH , [ U , V ]) ⊆ K j ( A H , [ U , V ]) for each j. Hence, if the block Lanczosprocess does not break down, the unitary matrix which transforms A H to block-tridiagonal form bringsalso A AH to block-tridiagonal form, and hence also A is brought to block tridiagonal form with blocks ofsize at most 4. Generalizations of these results to generic k -almost normal matrices with k ≥ In this section we specialize the previous results for the remarkable cases where A is a rank–one perturba-tion of a Hermitian or a unitary matrix. The case of perturbed Hermitian matrices is not directly coveredby Theorem 8. In fact, it assumes that rank ( ∆ ( A )) = rank ([ U , V ]) or, equivalently, that there exists a setof 2 k linearly independent vectors spanning the column space of ∆ ( A ) whenever A is k − almost normal.If A = H + xy H , where H is a Hermitian matrix, then it is easily seen that A is 2-almost normal and C = yx H − xy H . Generically, ∆ ( A ) has rank 4 but U = V and, therefore, rank ([ U , V ]) =
2. However, it isworth noticing that in this case C = UV H is antihermitian, i.e., C H = − C . By exploiting this additionalproperty of C we can prove that K j ( A AH , U ) ⊆ K ( A H , U ) , j ≥ , (3)meaning that the same unitary matrix which transforms the Hermitian part of A toi block tridiagonal form,also transforms the antihermitian part of A to block tridiagonal form and, therefore, A . In order to deduce(3), let us observe that ∆ ( A ) is Hermitian and CA − AC = A H C H − C H A H . Figure 1: Shape of the block tridiagonal matrix obtained from the block-Lanczos procedure applied to aarrow matrix with starting vectors in the column space of C Replacing C H = − C , we obtain that C ( A − A H ) = ( A − A H ) C , (4)meaning that the antihermitian part of A commutes with C . Multiplying both sides of (4) by the matrix V we have ( A − A H ) U = U ( V H ( A − A H ) V )( V H V ) − . which gives (3).Summing up, in the case of a rank–one modification of a Hermitian matrix we can apply the block-Lanczos procedure to A H starting with only two linearly independent vectors in the column space of C ,for example x and y , if known, thus computing a unitary matrix which transforms A to a block-tridiagonalmatrix with block size two. Differently, we can also employ a basis of S by obtaining a first block of size4 which immediately shrinks to size 2 in the subsequent steps. In Figure 1 and 2 we illustrate the shapesof the block tridiagonal matrices determined from the block-Lanczos procedure applied to an arrow matrixwith starting vectors in the column space of C and S , respectively.It is worth noticing that the matrix C plays an important role for the design of fast structured variantsof the QR iteration applied to perturbed Hermitian matrices. Specifically, in [4, 20] it is shown that thesequence of perturbations C k : = Q Hk C k − Q k yields a description of the upper rank structure of the matrices A k : = Q Hk A k − Q k generated under the QR process applied to A = A .The case where A = U + xy H is a rank–one correction of a unitary matrix U is particularly interestingfor applications to polynomial root-finding. If A is invertible, then it is easily seen that C = yx H + U xy H U H + y H U H x is such that A H A − CA = I n , AA H − AC = I n , which implies A H A − AA H = CA − AC . Thus, by applying Theorem 8 to the unitary plus rank–one matrix A we find that the block-Lanczos pro-cedure applied to A H starting with four linearly independent vectors in S reduces A to a block tridiagonalform as depicted in Figure 3. 7 Figure 2: Shape of the block tridiagonal matrix obtained from the block-Lanczos procedure applied to aarrow matrix with starting vectors in the column space of S Figure 3: Shape of the block tridiagonal matrix obtained from the block-Lanczos procedure applied to acompanion matrix 8he issue concerning the relationship between the matrices C k and A k generated under the QR iterationis more puzzling and, indeed, actually much of the work of fast companion eigensolvers is spent for theupdating of the rank structure in the upper triangular portion of A k . Moreover, if A = A is initially trans-formed to upper Hessenberg form by a unitary congruence then the rank of the off–diagonal blocks in theupper triangular part of A k is generally three whereas the rank of C k is two. Notwithstanding, the numericalbehavior of the QR iteration seems to be different if we apply the iterative process directly to the blocktridiagonal form of the matrix. In this case, under some mild assumption, it is verified that the rank of theblocks in the upper triangular portion of A k located out of the block tridiagonal profile is at most 2 and, inaddition, the rank structure of these blocks is completely specified by the matrix C k . The property of block tridiagonalization is inherited by a larger class of perturbed normal matrices [11]. Itis interesting to consider such extension even in simple cases in order to enlighten the specific features ofalmost normal matrices with respect to the band reduction and to the QR process. In this section we showthat the block-Lanczos procedure can be employed for the block tridiagonalization of rank–one perturba-tions of certain normal matrices whose eigenvalues lie on an algebraic curve of degree at most two. Thesematrices are not in general almost normal, but the particular distribution of the eigenvalues of the normalpart, guarantees the existence of a polyanalytic polynomial of small degree relating the antihermitian partof A with the Hermitian part of A .Let A ∈ C n × n be a matrix which can be decomposed as A = N + uv H , u , v ∈ C n , NN H − N H N = . (5)Also suppose that its eigenvalues λ j = ℜ ( λ j ) + i ℑ ( λ j ) , 1 ≤ j ≤ n , lie on a real algebraic curve of degree2, i.e., f ( ℜ ( λ j ) , ℑ ( λ j )) =
0, where f ( x , y ) = ax + by + cxy + dx + ey + f =
0. From ℜ ( λ ) = λ + ¯ λ , ℑ ( λ ) = λ − ¯ λ , by setting x = z + ¯ z , y = z − ¯ z , it follows that λ j , 1 ≤ j ≤ n , belong to an algebraic variety Γ = { z ∈ C : p ( z ) = } defined by p ( z ) = a , z + a , z ¯ z + a , ¯ z + a , z + a , ¯ z + a , = , with a k , j = ¯ a j , k . This also means that the polyanalytic polynomial p ( z ) annihilates N in the sense that p ( N ) = a , N + a , NN H + a , N H + a , N + a , N H + a , I n = . (6)If a , = ¯ a , = a , = Γ reduces to a , ¯ z = − ( a , z + a , ) . that is the case of a shifted Hermitian matrix N = H + γ I . As observed in the previous section rank–onecorrections of Hermitian matrices are almost-normal, and shifted almost normal matrices are almost-normalas well. Thus we can always suppose that the following condition named ( Hypothesis 1 ) is fulfilled a , + a , − a , (cid:54) = . (7)In fact when Hypothesis 1 is violated, but not all the terms above are zero, then we can consider the modifiedmatrix A (cid:48) = e i θ A = e i θ N + u (cid:48) v (cid:48) H and observe that the eigenvalues of e i θ N belongs to the algebraic variety a (cid:48) , z + a (cid:48) , z ¯ z + a (cid:48) , ¯ z + a (cid:48) , z + a (cid:48) , ¯ z + a (cid:48) , = , a (cid:48) , = a , / e θ , a (cid:48) , = a , / e − θ , a (cid:48) , = a , . Hence, for a suitable choice of θ it follows a (cid:48) , + a (cid:48) , − a (cid:48) , (cid:54) = . Under Hypothesis 1 it is easily seen that the leading part of p ( z ) can be represented in some usefuldiverse ways. In particular, the 3 × α , β and γ determined to satisfy α ( z − ¯ z ) z + β ( z + ¯ z ) z + γ ( z + ¯ z ) ¯ z = a , z + a , z ¯ z + a , ¯ z , (8)is given by γ = a , ; α + β = a , ; β + γ − α = a , . This system is solvable and, moreover, we have α = a , + a , − a , (cid:54) =
0. Analogously, the 3 × α , β and γ determined to satisfy α ( z − ¯ z ) ¯ z + β ( z + ¯ z ) z + γ ( z + ¯ z ) ¯ z = a , z + a , z ¯ z + a , ¯ z , (9)is given by β = a , ; γ − α = a , ; β + γ + α = a , . Again the system is solvable and α = − a , + a , − a , (cid:54) = A = N + uv H , with N normal matrix, the matrix ∆ ( A ) = A H A − AA H is a matrix of rank four atmost. Specifically, we find that ∆ ( A ) = A H uv H + vu H N − A vu H − uv H N H , and, hence, the space S is included in the subspace D : = span { u , v , A H u , A v } ⊆ D s : = span { u , v , A H u , A H v , A u , A v } . Also, recall that A AH · A H − A H · A AH = ∆ ( A ) . From this by induction it is easy to prove the following result, analogous to Lemma 1.
Lemma 9.
For any positive integer j we haveA AH · A jH = A jH · A AH + j − ∑ k = A kH · ∆ ( A ) · A j − − kH . If the procedure block-Lanczos applied to A H with initial matrix Z ∈ C n × (cid:96) , (cid:96) ≤ { Z } = D s terminates without premature stop then at the very end the unitary matrix U transforms A H into theHermitian block tridiagonal matrix T = U H · A H · U with blocks of size at most 6. The following result saysthat H : = U H · A AH · U is also block-tridiagonal with blocks of size at most 6. Theorem 10.
Let A ∈ C n × n be as in (5) , (6) and (7) . Then we have K j ( A AH , Z ) ⊆ K j ( A H , Z ) for eachj ≥ , whenever span { Z } = D s = span { u , v , A H u , A H v , A u , A v } . Hence, if the block Lanczos process doesnot break down, the unitary matrix which transform A H to block-tridiagonal form brings also A AH to block-tridiagonal form, and hence also A is brought to block tridiagonal form with blocks of size at most 6. roof. Let U ( : , i ) be the first block of columns of U spanning the subspace D s . The proof follows byinduction on j . Consider the initial step j =
1. We have A AH u = − A H u + A u ∈ K ( A H , U ( : , i ) and a similar relation holds for A AH v . Concerning A AH A u from (8) we obtain that α ( N − N H ) N + β ( N + N H ) N + γ ( N + N H ) N H = a , N + a , NN H + a , N H , and, hence, by using (6) we find that − α ( N − N H ) N = β ( N + N H ) N + γ ( N + N H ) N H + ( a , N + a , N H + a , I ) . (10)By plugging N = A − uv H into (10) we conclude that ( A − A H ) A u ∈ K ( A H , U ( : , i )) . We can proceed similarly to establish the same property for the remaining vectors A AH A v and A AH A H u , A AH A H v by using (9).To complete the proof, assume that A jAH U ( : , i ) ∈ K j ( A H , U ( : , i )) , and prove the same relation for j +
1. We have A j + AH U ( : , i ) = A AH ( A jAH U ( : , i ) = A AH X . By induction X belongs to K j ( A H , U ( : , i )) and, therefore, the thesis is proven by applying Lemma 9.Note that when we have a coefficient a i j = N is unitary, the polynomial becomes p ( z ) = z ¯ z − , meaning a , = a , = a , = a , = a , = a , = −
1. From (8) we have α = − / β = / γ =
0, and hence we can see that everything works starting from the vectors [ u , v , A u , A v ] independently ofthe invertibility of A as required in the previous section to establish the existence of a suitable matrix C .In general, however, four initial vectors are not sufficient to start with the block tridiagonal reductionsupporting the claim that for the given A there exist no matrix C of rank two satisfying ∆ ( A ) = CA − AC .However, due to the relations induced by the minimal polyanalytic polynomial of degree two it is seen thatthe construction immediately shrinks to size 4 after the first step. In figure 4 we show the shape of thematrix generated from the block-Lanczos procedure applied for the block tridiagonalization of a normal-plus-rank–one matrix where the normal component has eigenvalues located on some arc of parabola in thecomplex plane. In this paper we have addressed the problem of computing a block tridiagonal matrix unitarily similarto a given almost normal or perturbed normal matrix. A computationally appealing procedure relyingupon the block Lanczos method is proposed for this task. The application of the banded reduction for theacceleration of rank-structured matrix computations is an ongoing research topic.11
10 20 30 40 50 600102030405060 nz = 744
Figure 4: Shape of the block tridiagonal matrix obtained from the block-Lanczos procedure applied to arank–one correction of a normal matrix whose eigenvalues lie on some arc of parabola
References [1] R. Bevilacqua and G. M. Del Corso. A condensed representation of almost normal matrices.
LinearAlgebra and Its Applications , 438(11):4408–4425, 2013.[2] D. A. Bini, F. Daddi, and L. Gemignani. On the shifted QR iteration applied to companion matrices.
Electron. Trans. Numer. Anal. , 18:137–152 (electronic), 2004.[3] S. Delvaux and M. Van Barel. A Givens-weight representation for rank structured matrices.
SIAM J.Matrix Anal. Appl. , 29(4):1147–1170, 2007.[4] Y. Eidelman, L. Gemignani, and I. Gohberg. Efficient eigenvalue computation for quasiseparableHermitian matrices under low rank perturbations.
Numer. Algorithms , 47(3):253–273, 2008.[5] Y. Eidelman and I. Gohberg. On generators of quasiseparable finite block matrices.
Calcolo , 42(3-4):187–214, 2005.[6] L. Elsner and Kh. D. Ikramov. On a condensed form for normal matrices under finite sequences ofelementary unitary similarities.
Lin. Alg. its Appl. , 254:79–98, 1997.[7] L. Elsner and Kh. D. Ikramov. Normal matrices: an update.
Lin. Alg. and its Appl. , 285:291–303,1998.[8] V. Faber, J. Liesen, and P. Tich´y. On orthogonal reduction to hessenberg form with small bandwidth.
Numerical Algorithms , 51(2):133–142, 2009.[9] V. Faber and T. Manteuffel. Necessary and sufficient conditions for the existence of a conjugategradient method.
SIAM J. Numer. Anal. , 21(2):352–362, 1984.[10] K. Frederix, S. Delvaux, and M. Van Barel. An algorithm for computing the eigenvalues of blockcompanion matrices.
Numer. Algorithms , 62(2):261–287, 2013.[11] M. Gasemi Kamalvand and Kh. D. Ikramov. Low-rank perturbations of normal and conjugate-normalmatrices and their condensed forms with respect to unitary similarities and congruences.
VestnikMoskov. Univ. Ser. XV Vychisl. Mat. Kibernet. , (3):5–11, 56, 2009.[12] R. Grone, C. R. Johnosn, E. M. Sa, and H. Wolkowicz. Normal matrices.
Lin. Alg. and its Appl. ,87:213–225, 1987. 1213] M. H. Gutknecht and T. Schmelzer. Updating the QR decomposition of block tridiagonal and blockHessenberg matrices.
Appl. Numer. Math. , 58(6):871–883, 2008.[14] T. Huckle. The Arnoldi method for normal matrices.
SIAM J. Matrix Anal. Appl. , 15(2):479–489,1994.[15] M. Huhtanen. Orthogonal polyanalytic polynomials and normal matrices.
Math. Comp. ,72(241):355–373 (electronic), 2003.[16] Kh. D. Ikramov. On normal band matrices.
Zh. Vychisl. Mat. Mat. Fiz. , 37(1):3–6, 1997.[17] J. Liesen and P. E. Saylor. Orthogonal hessenberg reuction and orthogonal krylov subspace bases.
SIAM J. Numer. Anal. , 42(5):2148–2158, 2005.[18] M. Putinar. Linear analysis of quadrature domains. III.
J. Math. Anal. Appl. , 239(1):101–117, 1999.[19] R. Vandebril and G. M. Del Corso. An implicit multishift qr -algorithm for hermitian plus low rankmatrices. SIAM J. Sci. Comp. , 16(4):2190–2212, 2010.[20] R. Vandebril and G. M. Del Corso. A unification of unitary similarity transforms to compressedrepresentations.