Linear Algebra and its Applications 432 (2010) 3210–3230 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa A complete solution of the normal Hankel problem ∗ V.N. Chugunov , Kh.D. Ikramov Institute of Numerical Mathematics, Russian Academy of Sciences, ul.Gubkina 8, Moscow 119991, Russia Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie Gory, Moscow 119992, Russia ARTICLE INFO ABSTRACT Article history: The normal Hankel problem is the one of characterizing the matri- Received 15 June 2009 ces that are normal and Hankel at the same time. We give a complete Accepted 12 January 2010 solution of this problem. Available online 18 February 2010 © 2010 Elsevier Inc. All rights reserved. Submitted by R. Horn AMS classification: 15A21 Keywords: Hankel matrix Normal matrix Toeplitz matrix Backward identity Circulant Hankel circulant Separable circulant 1. Introduction The normal Toeplitz problem (NTP) is the one of characterizing the matrices that are normal and Toeplitz at the same time. This problem was posed and solved by the authors in [8,9,11]. (Other solutions of this problem were proposed in [1,5–7,18].) The normal Hankel problem (NHP) is the one of characterizing the matrices that are normal and Hankel at the same time. It turned out to be much harder than the NTP and was open for many years (see [2,3,10,12–17]). In this paper, we give a complete solution of this problem. ∗ Corresponding author. Address: Institute of Numerical Mathematics, Russian Academy of Sciences, ul.Gubkina 8, Moscow 119991, Russia. E-mail address: [email protected] (V.N. Chugunov). 0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.01.021 V.N. Chugunov, Kh.D. Ikramov / Linear Algebra and its Applications 432 (2010) 3210–3230 3211 Let NHn be the set of normal Hankel matrices of order n. With each matrix H ∈ NHn, we associate the Toeplitz matrix T = HPn, (1) where ⎛ ⎞ 1 ⎝ ... ⎠ Pn = 1 is the backward identity matrix of order n. One can easily verify the following proposition: ∗ Proposition 1. A Hankel matrix H is normal if and only if the matrix TT is real; that is, ∗ Im TT = 0. (2) Proposition 1 implies that, instead of characterizing NHn, we may describe the corresponding Toeplitz matrices. Suppose that matrix (1) is written in the algebraic form T = T1 + iT2, (3) where T + T T − T T1 = ,T2 = . (4) 2 2i As usual, the bar over the symbol of a matrix or a vector denotes the entry-wise complex conjugation. Substituting (3) into (2), we obtain yet another normality condition for the original matrix H. Proposition 2. A Hankel matrix H is normal if and only if t = t . T1T2 T2T1 (5) Let a1, ...,an−1 and a−1, ...,a−n+1 be the off-diagonal entries in the first row and the first column of T1. Similarly, let b1, ...,bn−1 and b−1, ...,b−n+1 be the off-diagonal entries in the first row and the first column of T2. Form the matrices ⎛ ⎞ an−1 bn−1 ⎜ ⎟ ⎜an−2 bn−2⎟ F = ⎜ . ⎟ (6) ⎝ . ⎠ a1 b1 and ⎛ ⎞ a−1 b−1 ⎜ ⎟ ⎜ a−2 b−2 ⎟ G = ⎜ . ⎟ . (7) ⎝ . ⎠ a−n+1 b−n+1 In Section 2, we show that, if both matrices F and G are rank-deficient, then H must belong to one of the following four classes: 1. Arbitrary complex multiples of real Hankel matrices. 2. Matrices of the form αPn + βH, α, β ∈ C, where H is an arbitrary real centrosymmetric Hankel matrix. 3212 V.N. Chugunov, Kh.D. Ikramov / Linear Algebra and its Applications 432 (2010) 3210–3230 3. Block diagonal matrices of the form αH1 ⊕ βH2, α, β ∈ C, where H1 is a real upper triangular Hankel matrix of order k (with 0 < k < n) and H2 is a real lower triangular Hankel matrix of order l = n − k. We call H1 and H2 an upper triangular and a lower triangular Hankel matrix, respectively, if {H1}ij = 0fori + j > k + 1 and {H2}ij = 0fori + j < l + 1. 4. Matrices of the form − αH + βH 1, α, β ∈ C, where H a nonsingular real upper triangular (or lower triangular) Hankel matrix. Now, assume that at least one of the matrices F and G has full rank. In Section 3, we show that, in this case, both F and G have rank two and obey the relation G = FW (8) for some real 2 × 2 matrix αβ W = γδ, (9) with a unit determinant: αδ − βγ = 1. (10) In view of definitions (6) and (7), matrix equality (8) is equivalent to the scalar relations a−i = αan−i + γ bn−i,b−i = βan−i + δbn−i, 1 i n − 1. (11) Writing the Toeplitz matrix (1) in the form ⎛ ⎞ t0 t1 t2 ... tn−1 ⎜ ⎟ ⎜ t−1 t0 t1 ... tn−2⎟ ⎜ ⎟ T = ⎜ t−2 t−1 t0 ... tn−3⎟ , (12) ⎝ ... ... ... ... ...⎠ t−n+1 t−n+2 t−n+3 ... t0 we can replace real relations (11) with the complex formula t−i = φtn−i + ψtn−i, 1 i n − 1, (13) where α + δ β − γ α − δ β + γ φ = + i , ψ = + i . (14) 2 2 2 2 Then, relation (10) takes the complex form |φ|2 −|ψ|2 = 1. (15) The case ψ = 0, |φ|=1 corresponds to the well-known class of φ-circulants. For this reason, matrices defined by relation (13) for a fixed pair (φ, ψ) were called (φ, ψ)-circulants in [14]. Thus, beginning from Section 4, we deal only with various classes of (φ, ψ)-circulants. Each of these classes is specified by the corresponding matrix W (see (9)). In Section 4, we prove an important lemma that shows that the case of a general matrix W obeying relation (10) can be reduced to W having diagonal or Jordan form. Consequently, the following four cases must be distinguished: V.N. Chugunov, Kh.D. Ikramov / Linear Algebra and its Applications 432 (2010) 3210–3230 3213 1. The eigenvalues λ1 and λ2 of W are complex conjugate. 2. The eigenvalues λ1 and λ2 are real and distinct. 3. The eigenvalues λ1 and λ2 are identical, and W is diagonalizable. 4. The eigenvalues λ1 and λ2 are identical, and the Jordan form of W is a Jordan block of order two. These four cases are dealt with in Sections 5–8. 2. Rank-deficient case Suppose that one of the matrices T1 and T2 in (3) is diagonal and nonzero. For definiteness, assume that T1 = αIn. Then, relation (5) says that T2 is a (real) symmetric Toeplitz matrix. The corresponding matrix H = TPn has the form H = αPn + iT2Pn and, hence, belongs to class 2. Therefore, in what follows, we assume that neither T1 nor T2 is diagonal. It will be convenient to isolate the diagonal parts in both T1 and T2: T1 = a0In + T1,T2 = b0In + T2. (16) Here, T1 and T2 have the zero principal diagonal. Substituting (16) into (5), we obtain t − t = (t − ) − (t − ). T2T1 T1T2 a0 T2 T2 b0 T1 T1 (17) This equation characterizes all the matrices T corresponding to the matrices in NHn. In particular, (17) implies that, for every such T, the matrix t − t T2T1 T1T2 must be Toeplitz. Let us discuss the consequences of this fact. From the equalities ( t − t ) = ( t − t ) = ... − T2T1 T1T2 i+1,j+1 T2T1 T1T2 i,j,i,j 1, ,n 1, we derive n n n n (T2)i+1,k(T1)j+1,k − (T1)i+1,k(T2)j+1,k − (T2)ik(T1)jk + (T1)ik(T2)jk = 0 k=1 k=1 k=1 k=1 or n n n n bk−i−1ak−j−1 − ak−i−1bk−j−1 − bk−iak−j + ak−ibk−j = 0, k=1 k=1 k=1 k=1 i, j = 1, ...,n− 1. We change here the summation indices; namely, we set m = k − 1 in the first and second sums and m = k in the third and fourth sums. This yields n−1 n−1 n n bm−iam−j − am−ibm−j − bm−iam−j + am−ibm−j = 0, m=0 m=0 m=1 m=1 whence an−ibn−j − an−jbn−i = a−ib−j − a−jb−i,i,j= 1, ...,n− 1. (18) 3214 V.N. Chugunov, Kh.D. Ikramov / Linear Algebra and its Applications 432 (2010) 3210–3230 Define ΔF = an−i bn−i = − ij det an−ibn−j an−jbn−i an−j bn−j and ΔG = a−i b−i = − . ij det a−ib−j a−jb−i a−j b−j Now, relation (18) take the form ΔF = ΔG = ... − . ij ij ,i,j 1, ,n 1 (19) So far, the ranks of the matrices F and G were not important. Now, for the rest of this section, we assume that rank F < 2 and rank G < 2. The case rank F = rank G = 0 is clearly impossible since, otherwise, T1 and T2 would be diagonal matrices. For the other values of rankF and rankG we give separate analyses. t It will be convenient to use the following notation. For a vector f = (f1,f2, ...,fn−1) , the upper triangular Toeplitz matrix with the first row (0 f1 f2 ... fn−1) is denoted by T (f ). The symbols u1 and u2 stand for the columns of F, while the columns of G are denoted by l1 and l2. 2.1. F =/ 0,G= 0 ΔG = ∀ ΔF = = In this case, we have ij 0 i, j.Inviewof(19), ij 0 for all i, j; hence, rank F 1.