On some reciprocal matrices with elliptical components of their Kippenhahn curves


 By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai
 
 ,
 
 i
 
 +1
 ai
 
 +1,
 
 i
 = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.


Introduction
Denote by Mn the algebra of all n-by-n matrices with complex entries. For any A ∈ Mn, we will use the standard notation Re Note that Λ (A) is the regular numerical range W(A) of A, and the respective formula (1.3) is an immediate consequence of its convexity (the celebrated Toeplitz-Hausdor theorem, see e.g. [7]). For k > , however, (1.3) is a more recent result [10]. It implies the convexity of all Λ k (A) -the result conjectured in [5,Problem 2.9] and established independently in [11].
Due to (1.3), all the sets Λ k (A) are de ned completely by the envelope C(A) of the family (1.2). In particular, W(A) is the convex hull of C(A), as was observed as early as in [8] (see [9] for the English translation). Called the "boundary generating curve" in [8], in the recent years C(A) was renamed as Kippenhahn curve -the term which we will be using in what follows.
Along the same lines (no pun intended), the family (1.2) is de ned by (1.1), and thus by the characteristic polynomial P(λ, θ) = det(Re(e iθ A) − λI) (1.6) of Re(e iθ A); the latter is therefore called the Kippenhahn polynomial of A. A vast number of papers is devoted to singling out classes of matrices A with W(A) being an elliptical disk, the convex hull of several ellipses, or at least having an elliptical arc as part of its boundary ∂W(A). Similar questions can be asked about higher rank numerical ranges. In terms of Kippenhahn curves, this boils down to the following: when does C(A) contain an ellipse E as one of its components (or, more restrictively, consists of several ellipses)?
In [2] this question was addressed for so called reciprocal matrices of small (up to n = ) sizes, provided that the ellipse E is centered at the origin. This condition on E is lifted here. Also, the higher rank numerical ranges of the respective matrices A are described.

General observations
The classical Elliptical Range Theorem claims that the numerical range W(A) of A ∈ M is an elliptical disk with the foci at the eigenvalues ζ , ζ of A and the minor axis of the length c : The third and the fourth equations of (2.2) imply that x + y = |ζ − ζ | and so, invoking the fth: In particular, Proposition 1 is not new; its particular case corresponding to the ellipticity criterion for W(A) in terms of an explicit formula for λ (θ) is [4,Theorem 1]. In slightly di erent notation, and without specifying the minor axis and foci of E, it was also used in [6]. We chose to state it in a detailed form for convenience of reference. Following [2], we say that a matrix A = (a ij ) n i,j= is reciprocal if it is tridiagonal with constant main diagonal, i.e. a ij = whenever |i − j| > , while a = . . . = ann := a and a i,i+ a i+ ,i = for i = , . . . , n − . (2.5) In what follows, we will denote by RMn the class of all reciprocal matrices A ∈ Mn with a = . The latter condition is just a technicality, and it is being imposed only for convenience. Propositions 4,8, Lemmas 1,2, Theorems 5,7, and Corollaries 1,3,4 hold exactly as stated for any value of a. All other results of Sections 2-4 can easily be adjusted to arbitrary a based on the simple observation C(A + cI) = C(A) + c.
For reciprocal A, all the matrices Re(e iθ A) are also tridiagonal with a constant main diagonal. This observation was used in [2,Proposition 4] to conclude that Kippenhahn polynomials of A ∈ RMn can be written as premultiplied by −λ if n is odd. Here ζ = λ , k = n/ , and p j are polynomias of degree k − j in τ := cos( θ) with the coe cients depending only on In this paper, we nd it more convenient to represent Pn as with the variable τ replaced by ρ := cos θ = ( + τ)/ and the coe cients of p j expressed in terms of Note that ξ j ≥ , and the equality holds if and only if a j,j+ (equivalently, A j , or a j+ ,j ) is equal to one.

Proposition 2. Suppose the Kippenhahn curve C(A) of a reciprocal matrix A contains an ellipse E. Then
(i) the foci of E are real, and (ii) either E is centered at the origin, or its re ection −E is also contained in C(A).
Proof. Part (ii) follows immediately from the symmetry of C(A). To prove (i), recall that for any square matrix the foci of its Kippenhahn curve coincide with its spectrum [9,Theorem 11]. If A is reciprocal, it is similar (not unitarily similar!) to the Toeplitz tridiagonal matrix T, as was observed in [2, Proposition 5] by the anonymous referee's suggestion. Consequently, for any A ∈ RMn its spectrum is real. The foci of E lie in the set (2.9), and as such are also real.
Proof. Indeed, Λ k (A) for k > n/ can only be a singleton coinciding with an eigenvalue of A of multiplicity at least k − n [5, Proposition 2.2], and (2.9) implies that all the eigenvalues of A are simple.
According to Proposition 2(i), for an ellipse E ⊂ C(A) in case of reciprocal A we have q = y = , with the rest of formulas (2.2) thus simplifying to Proposition 1 for reciprocal matrices can therefore be recast as follows.

Proposition 3. Let A ∈ RMn. Then C(A) contains an ellipse with the half-axes of length c and
Case (a) corresponds to an ellipse centered at the origin. In case (b) there are two symmetric ellipses ±E contained in C(A) by Proposition 2(ii); ±p denote their centers. Formula (2.11) is obtained by multiplying out the respective polynomials P ±E = (λ ∓ p cos θ) − (x cos( θ) + z); in both cases we relabeled x =: X . There is another important consequence of Re(e iθ A) being tridiagonal hermitian matrices. According to [3,Corollary 7], such matrices can have a repeated eigenvalue only if they split into the direct sum of two blocks sharing this eigenvalue. Due to (2.5), in our setting this can only happen if θ = π/ mod π and A k = (equivalently: ξ k = ) for some k.

Proposition 4. For a reciprocal matrix A, the multiple tangent lines of C(A) can only be horizontal. Such lines materialize if and only if ξ k = for some k, and the spectra of the left upper k-by-k and the right lower (n − k)by-(n − k) block of the matrix K = Im A overlap.
Proof. We just need to recall that tangent lines to C(A) form the family (1.2).
Due to the symmetry of C(A), its multiple tangent lines, if any, come in pairs symetric with respect to the abscissa axis. Their ordinates are the multiple eigenvalues of Im A.
Note that the presence of an ellipse E ⊂ C(A) centered at p ≠ implies the existence of lines tangent to both E and −E. Proposition 4 thus implies Corollary 2. Let A ∈ RMn be such that C(A) contains an ellipse E with its center di erent from the origin. Then (i) E ∩ (−E) ≠ ∅ and (ii) a k,k+ = (equivalently: ξ k = ) for some k ∈ { , . . . , n − }, while the upper left k-by-k block of Im A has a non-zero eigenvalue in common with its lower right (n − k)-by-(n − k) block.
Proof. Being congruent, the ellipses E and −E either intersect or lie outside of each other. In the latter case, they would have a non-horizontal tangent in common which would contradict Proposition 4. This proves (i).
Part (ii) also follows from Proposition 4 as soon as we observe that a shared eigenvalue equal to zero corresponds to E degenerating into the pair of its foci. If k = or n − , one of the blocks of K is a onedimensional { } and, as such, cannot generate a non-zero eigenvalue.
In contrast with Corollary 2(i), the concentric elliptical components of C(A), if any, have to be nested.

Corollary 3. Suppose the Kippenhahn curve of a reciprocal matrix A contains two concentric ellipses. Then one of them has to lie inside the other.
Proof. The reason is exactly the same as in the proof of the previous result: the absence of non-horizontal multiple tangents of C(A).
In conclusion, note that reversing the order of rows and columns of a matrix is a unitary similarity which preserves the reciprocal structure, while switching the super-and subdiagonal and reversing the order of their entries. Therefore, the Kippenhahn polynomials and curves of reciprocal matrices are invariant under the transformation ξ j ←→ ξ n−j+ , j = , . . . , n − . (2.12) This simple observation will prove itself useful in the next sections.

Reciprocal 4-by-4 and 5-by-5 matrices
Let us start with A ∈ M . The Kippenhahn polynomial (1.6) of A then has degree four in λ. So, if a quadratic polynomial can be factored out, the remaining multiple is also quadratic. In the language of Kippenhahn curves it means that if C(A) contains an elliptical component, it actually consists of two ellipses. If A ∈ RM , Proposition 2(ii) implies in addition that the two ellipses in question are either both centered at the origin, or are re ections of each other through the origin (equivalently: across the ordinate axis).
According to (2.9) with n = : where ϕ = ( √ + )/ is the golden ratio, while (2.7) takes the form In particular, the characteristic polynomial of Im A is

4)
with at least one of ξ j being di erent from zero, or Case (3.4) corresponds to ellipses centered at the origin, and the criterion (up to the notational change from A j to ξ j ) is [2,Theorem 8]. According to (3.1), one of the ellipses (say, E ) has foci ±ϕ while the foci of E then are ±ϕ − . The lengths of the minor half-axes are determined by the positive roots of (3.3), and direct computations yield ξ ϕ, ξ ϕ − or ξ ϕ, ξ ϕ − , depending on which of the equalities holds in (3.4). Note that the situation falls under (3.4) and is formally treated as two concentric ellipses, in spite of the fact that the inner one degenerates into the pair of its foci. Also, under condition (3.4) at least one of ξ j being di erent from zero implies that only one of them can actually equal zero. Criterion (3.5) (once again, up to the notational change from A j to ξ j ) was derived in [6, Theorem 6.1] as a corollary of a more general Theorem 5.1 on 4-by-4 matrices with scalar diagonal blocks. Here is a streamlined reasoning, speci c for the case at hand.
Necessity. Suppose C(A) = E ∪ E with E , being symmetric images of each other. According to Corollary 2, ξ = and σ(Im A) consists of two opposite eigenvalues, each of multiplicity 2. From (3.3) we conclude that ξ = ξ .
Su ciency. Let (3.5) hold. Denoting the common value of ξ , by c , it is easy to see that (3.2) can be rewritten as (2.11) with p = / and X = √ / . Proposition 3 then implies that E and E are congruent ellipses with the foci ϕ, −ϕ − and −ϕ, ϕ − , respectively, and c as the length of their minor half-axes. (3.5). These results are also from [2] and [6], respectively.

Theorem 5 immediately implies that W(A) is the elliptical disk bounded by E if (3.4) holds, and conv{E ∪ E } under condition
Formulas (1.3) allow us to move further along the chain (1.5).

Corollary 4. Let A ∈ RM satisfy (3.4) or (3.5). Then Λ (A) is the elliptical disk bounded by the inner ellipse E in case (3.4), and the intersection of the elliptical disks bounded by E and E in case (3.5).
Note that Λ We now move to n = . The Kippenhahn polynomial of A ∈ RM is a product of −λ by Qualitatevely, (3.7) is similar to (3.2), implying that in our current setting again C(A) contains two ellipses as soon as it contains at least one. Moreover, these ellipses are either both centered at the origin (which for n = is the third component of C(A)), or are re ections of each other through the ordinate axis. The foci of these ellipses are located at the non-zero points of σ(A), which are ± √ , ± . According to [2, Theorem 9], for two concentric ellipses to materialize it is necessary and su cient that with not all of ξ j equal zero. Note that due to (3.8) at least two of ξ j are then di erent from zero. Computing the roots of (3.7) with ρ = , we nd that under either of equalities (3.8) the lengths of the minor half-axes of the ellipses are (ξ + ξ )/ and ξ + ξ + (ξ + ξ )/ .
By Corollary 3, they correspond to the ellipses with their foci at ± and ± √ , respectively. The inner ellipse degenerates if and only if ξ = ξ = .
Passing to the case of displaced ellipses, let us rst derive the criterion for C(A) to have multiple tangent lines.

Lemma 1. A matrix A ∈ RM has the Kippenhahn curve with multiple tangent lines if and only if
(3.9) Proof. By Proposition 4, only horizontal multiple tangent line are possible, and ξ ξ = is a necessary condition for them to materialize. Due to the invariance of (3.9) under (2.12), we may without loss of generality suppose that ξ = and concentrate on showing that then is the desired criterion.
The remaining part of the requirements of Proposition 4 is that the spectra of B and B overlap; in other words, |η | = |η | + |η | . Since η j = ξ j , the necessity is proven. The case ξ = has to be excluded, because otherwise C(A) degenerates into σ(A).
The su ciency can be demonstrated via a straightforward computation. Since the blocks B , B are adjacent, it also follows from the result of [3, Theorem 10] applicable to arbitrary tridiagonal matrices.
Note that the multiple eigenvalues of Im A are in fact its maximal and minimal ones. So, the horizontal tangent lines of C(A) are the supporting lines of W(A). Lemma 1 therefore delivers the criterion for the numerical range of a reciprocal 5-by-5 matrix to have at portions on its boundary. Finally, observe that (3.10) may hold while (3.12) does not. This means that the numerical range W(A) of a reciprocal 5-by-5 matrix may have at portions on its boundary while the smooth arcs connecting these (horizontal) at portions are not elliptical. The proof of the criterion (3.5) outlined on p. 8 shows that for n = this is an impossibility.

Theorem 6. The Kippenhahn curve of A ∈ RM contains two non-concentric ellipses if and only if
To illustrate, consider A ∈ RM with ξ = ξ = . , ξ = ξ = .  Observe also that under condition (3.17) A is what in [1] was called a 2-periodic reciprocal matrix. Such matrices of dimension n = mod (in particular, n = ) have no elliptical components in C(A) according to [1,Theorem 5].
To conclude this section, here is an example of C(A) for a matrix A satisfying conditions of Theorem 6. It consist of the origin and a pair of displaced ellipses. The rank-2 numerical range is the intersection of the two elliptical discs.

Reciprocal 6-by-6 matrices
The Kippenhahn polynomial of A ∈ RM is In contrast with the n = , cases, Proposition 1 implies that C(A) may contain both elliptical and nonelliptical components. When this happens, the ellipse contained in C(A) is necessarily centered at the origin, and the respective criterion is delivered by [2,Theorem 10]. On the other hand, if C(A) contains two ellipses, then it is in fact the union of three ellipses, either all centered at the origin, or one centered at the origin and two symmetric through the ordinate axis. The former case is described by [2,Theorem 11], and in our notation (after some additional manipulations) can be restated as follows.

Theorem 7. Let A ∈ RM . Then for C(A) to consist of three concentric ellipses it is necessary and su cient that
where ξ j are de ned by (2.8).
With conditions (4.2) satis ed, C(A) = ∪ j= E j , where according to (2.9) E j is an ellipse with the foci ± cos jπ . The lengths of the minor half-axes of E j are the positive eigenvalues of Im A, i.e., positive roots c j of Proof. Let A(t) be a reciprocal matrix obtained from A by replacing its parameters ξ i with tξ i (t > ), i = , . . . , . Since conditions (4.2) are homogeneous, all the matrices A(t) will satisfy them along with A, and so C(A(t)) = ∪ j= E j (t). Moreover, the polynomial (4.3) is homogeneous in λ , ξ j , and so the multiset {c j (t)} j= of its roots is simply { √ tc j } j= . The foci of ellipses E j (t) do not depend on t while their minor axes depend on √ t linearly. Observe that the line segment I j := [− cos jπ , cos jπ ] connecting the foci of E j (t) lies inside it. On the other hand, E j (t) lies in an arbitrarily small neighborhood of I j provided that t is chosen small enough. So, for such t the ellipse E (t) lies inside E (t) which, in turn, lies inside E (t). Consequently, c (t) ≥ c (t) ≥ c (t) for small t. But then the inequalities persist for all t, since the ellipses E j (t) remain nested for all t by Corollary 3, and so their ordering cannot change. Setting t = completes the proof. (4.2). Denote by E j the ellipse with the foci ± cos jπ and the length of its minor half-axis equal the jth (in the non-increasing order) eigenvalue of Im A, j = , , . Then Λ j (A) is the elliptical disk bounded by E j .

Corollary 6. Let A be a reciprocal matrix satisfying
We now turn to the case of non-concentric ellipses. As in Section 3, we rst need to gure out when C(A) admits multiple tangent lines. (iv) ξ = , ξ ξ ξ ξ ≠ , and ξ ξ = (ξ − ξ )(ξ − ξ ). (4.5) Proof. By Proposition 4, for multiple tangent lines to exist it is necessary that at least one ξ j is equal to zero. So, we just need to consider each of the following situations separately. Case 1. ξ ξ ξ = . Then Im A splits into the direct sum of at least two diagonal blocks each of which is singular. So, is a multiple eigenvalue, and the abscissa axis is a multiple tangent line. No additional conditions on ξ j arise in this case.
Proof. By Proposition 3, where E (resp., E) is an ellipse centered at the origin (resp., some p > ) if and only if the polynomial (4.1) factors as (4.10) Recall also that X and c (X and c) are half the distance between the foci and half the length of the minor axis of E (resp. E).
Equating the respective coe cients of (4.1) and (4.10), we arrive at the system of nine equations which for our purposes is convenient to group into three subsystems (4.11)-(4.13), with three equations in each:  } Based on the above description of C(A), we arrive at the following result concerning the rank-j numerical ranges of A for j = , , (note that Λ j (A) = ∅ for j ≥ due to Corollary 1).

Corollary 7.
In the setting of of Theorem 9, the rank-j numerical ranges of A are as described by the following table: (4.6) or (4.7), (4.8) (4.7), (4.8) with k = cos π with k = cos π conv Note that under condition (4.6) conv E is the line segment I while under (4.7) or (4.8) it is a non-degenerate elliptical disk.