The diameter and width of numerical ranges

Linear Algebra and its Applications 582 (2019) 76–98

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Linear Algebra and its Applications www.elsevier.com/locate/laa

The diameter and width of numerical ranges Mao-Ting Chien a,∗ , Hiroshi Nakazato b , Jie Meng c a

Department of Mathematics, Soochow University, Taipei 11102, Taiwan Department of Mathematics and Physics, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan c Finance · Fishery · Manufacture Industrial Mathematics Center on Big Data, Pusan National University, Busan 46241, South Korea b

a r t i c l e

i n f o

Article history: Received 5 March 2019 Accepted 30 July 2019 Available online 5 August 2019 Submitted by C.-K. Li MSC: 15A60 52A10

a b s t r a c t We characterize the diameter and width of the numerical range, present an algorithm for computing the diameter and width of the numerical range, and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions. We also determine the boundaries of the numerical ranges of certain nilpotent Toeplitz matrices to be curves of constant width. © 2019 Elsevier Inc. All rights reserved.

Keywords: Diameter Width Constant width Numerical range Unitary bordering matrices Nilpotent Toeplitz matrices

* Corresponding author. E-mail addresses: [email protected] (M.-T. Chien), [email protected] (H. Nakazato), [email protected] (J. Meng). https://doi.org/10.1016/j.laa.2019.07.036 0024-3795/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction Let A be an n × n matrix. The numerical range of A is defined and denoted by W (A) = {Aξ, ξ : ξ ∈ C n , ξ, ξ = 1}. This set is convex due to Toeplitz-Hausdorff theorem (cf. [10,12]). The boundary ∂W (A) lies on an algebraic curve of degree less than or equal to n(n − 1)/2. Kippenhahn [12] characterized that W (A) is the convex hull of the real affine part of the dual curve of the algebraic curve FA (t, x, y) = 0, where the associated ternary form FA (t, x, y) = det(tIn + x(A) + y(A)), and (A) = (A + A∗ )/2, (A) = (A − A∗ )/(2i). Mirsky [15] introduced the concept of the spread of A which is defined as s(A) = sup{|λi − λj | : λ1 , . . . , λn are eigenvalues of A}. Several authors, for instance [11,15,20], obtained lower and upper bounds of the spread of A. In particular, when A is Hermitian, s(A) = max eigenvalue−min eigenvalue of A. Let Γ be a compact convex set of the Gaussian plane C. The diameter diam(Γ) of the set Γ is defined to be the largest distance of two parallel lines tangent to its boundary, and the width width(Γ) the smallest distance of two parallel lines tangent to its boundary. The boundary curve of a compact convex set Γ is called a curve of constant width if diam(Γ) = width(Γ). Clearly, the diameter of the convex hull of the spectrum of a matrix is precisely equal to the spread of the matrix. Tsing [21] gave a characterization of the diameter and width of the numerical range, and Bourin and Mhanna [1] used the width of the numerical range to estimate the symmetric norm of block-matrices. Rabinowitz [19] found parametric equations of polynomial equations whose graphs are non-circular curves of constant width. In this paper, we investigate the diameter and width of the numerical range of a matrix, provide an algorithm for computing the diameter and width of the numerical range, formulate the diameter of the numerical range of some unitary bordering matrices, and determine the condition for the boundary of the numerical range of certain Toeplitz matrices to be a curve of constant width. 2. Computing the diameter and width Let A be an n × n matrix. For 0 ≤ θ ≤ 2π, we consider the Cartesian decomposition e−iθ A =

e−iθ A − eiθ A∗ e−iθ A + eiθ A∗ +i = (e−iθ A) + i(e−iθ A). 2 2i

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We also use HA (θ) for the Hermitian matrix (e−iθ A). The n eigenvalues of HA (θ) are denoted by λ1 (θ) ≥ λ2 (θ) ≥ · · · ≥ λn (θ). Moreover, it is known that the numerical range W ((e−iθ A)) is the closed interval [λn (θ), λ1 (θ)]. Accordingly, we have the following observation of the diameter and width of the numerical range. Theorem 2.1. Let A be an n × n matrix. Then diam(W (A)) = max{λ1 (θ) − λn (θ) : 0 ≤ θ ≤ 2π}

(2.1)

width(W (A)) = min{λ1 (θ) − λn (θ) : 0 ≤ θ ≤ 2π},

(2.2)

and

where λ1 (θ) ≥ λ2 (θ) ≥ · · · ≥ λn (θ) are eigenvalues of HA (θ). To provide an algorithm for computing the diameter and width of the numerical range, we apply the resultant of two polynomials due to Sylvester (cf. [22]) which is defined as follows. Let f (Y ) = am Y m + am−1 Y m−1 + · · · + a1 Y + a0 and g(Y ) = bn Y n + bn−1 Y n−1 + · · · + b1 Y + b0 be two polynomials in Y with non-zero leading coefficients am , bn , and the coefficients aj , bk are functions in some other variables. The resultant of f and g with respect to Y is defined as the determinant of the (n + m) × (n + m)-matrix: ⎛

am ⎜ 0 ⎜ . ⎜ . ⎜ . ⎜ 0 ⎜ ⎜ 0 R=⎜ ⎜ bn ⎜ ⎜ 0 ⎜ . ⎜ . ⎜ . ⎝ 0 0

am−1 am .. . 0 0 bn−1 bn .. . 0 0

am−2 am−1 .. . 0 0 bn−2 bn−1 .. . 0 0

... 0 ... 0 .. .. . . . . . a1 . . . a2 ... 0 ... 0 .. .. . . . . . b1 . . . b2

0 0 .. . a0 a1 0 0 .. . b0 b1

⎞ 0 0⎟ .. ⎟ ⎟ . ⎟ 0⎟ ⎟ a0 ⎟ ⎟. 0⎟ ⎟ 0⎟ .. ⎟ ⎟ . ⎟ 0⎠ b0

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It is well known f and g have a common zero in Y if and only if det(R) = 0. Given θ ∈ [0, 2π], e−iθ A = (cos θ − i sin θ)((A) + i(A)), it follows that HA (θ) = cos θ (A) + sin θ (A). Hence, the characteristic polynomial of HA (θ) is exactly equal to FA (t, − cos θ, − sin θ) = det(tI − cos θ (A) − sin θ (A)). The following result characterizes the diameter and width of the numerical range. Theorem 2.2. Let A be an n × n matrix and let R(Z, x, y) be the resultant of FA (Y, x, y) and FA (Y + Z, x, y) with respect to Y . Denote KA (Z, x, y) = R(Z, x, y)/Z n . Then diam(W (A)) = max{Z ∈ R : KA (Z, − cos θ, − sin θ) = 0, 0 ≤ θ ≤ 2π}, and width(W (A)) = min max{Z ∈ R : KA (Z, − cos θ, − sin θ) = 0}. 0≤θ≤2π

Proof. Our strategy is applying the resultant induced by the polynomial FA (t, − cos θ, − sin θ) in t to characterize the diameter and width of the numerical range. Obviously, the function λ1 (θ) − λn (θ) is a zero of the polynomial 

(Z − (λj (θ) − λk (θ))).

1≤j=k≤n

This means that once we obtained the above polynomial, the diam(W (A)) according to Theorem 2.1, can be derived. Hence, we have to construct a product polynomial 

(Z − (λj − λk ))

1≤j=k≤n

from the polynomial p(t) = tn + a1 tn−1 + · · · + an−1 t + an =



(t − λi ).

1≤i≤n

For this reason, we assume that R(Z) is the resultant of p(Z + Y ) and p(Y ) with respect to Y . Then R(Z)/Z n is the required polynomial:

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(Z − (λj − λk )).

1≤j=k≤n

Accordingly, we consider the resultant R(Z, − cos θ, − sin θ) of the polynomials FA (Y, − cos θ, − sin θ) and FA (Y + Z, − cos θ, − sin θ) with respect to Y . We obtain that the quotient R(Z, − cos θ, − sin θ)/Z n is K(Z, − cos θ, − sin θ) =



(Z − (λj (θ) − λk (θ))),

1≤j=k≤n

and thus the formulae of diam(W (A)) and width(W (A)) follow. 2 The characterization of Theorem 2.1 provides a numerical algorithm to compute the diameter diam(W (A)) and the width width(W (A)) for an n × n matrix A: The value λ1 (HA (θ)) −λn (HA (θ)) in formulae (2.1) and (2.2) depends continuously on the angle 0 ≤ θ ≤ π. Let M be a sufficiently large natural number, and θ = mπ/M . We compute numerically the eigenvalues of the Hermitian matrix HA (θ). Then the diameter and width of W (A) can be approximated respectively by max{λ1 (HA (mπ/M )) − λn (HA (mπ/M )) : m = 0, 1, 2, . . . , M }, min{λ1 (HA (mπ/M )) − λn (HA (mπ/M )) : m = 0, 1, 2, . . . , M }. Concerning computational errors, we take the real part of the n approximate eigenvalues of HA (θ). We may use, for instance, Mathematica command “Sort” to arrange a real finite sequence s in decreasing order by doing “-Sort[-s]”. For each angle θ = mπ/M , we produce the value vm = λ1 (HA (mπ/M )) −λn (HA (mπ/M )). The command “ListPlot” graphs the points Pm = (vm , m), and “Max” and “Min” for the list of real numbers vm . We obtain the approximated values of diam(W (A)) and width(W (A)), which gives a numerical algorithm for computing diameter and width of the numerical range. The error bound of this method would be an interesting issue. The polynomial K(Z, − cos θ, − sin θ) has degree N = n(n − 1) with coefficients depending on θ. We may apply the numerical computation method used in [18, Appendix K] to obtain a priori error bound. On the other hand, we propose an analytic algorithm to express the values diam(W (A)) and width(W (A)) as the roots of an algebraic equation. This method is rather efficient if n is small. Changing the variable s = tan(θ/2) in the homogeneous polynomial KA (Z, x, y) in Theorem 2.2, the two quantities diam(W (A)) and width(W (A)) are formulated by

1 − s2 2s diam(W (A)) = sup Z ∈ R : KA (Z, − , − ) = 0, −∞ < s < ∞ , 1 + s2 1 + s2 1 − s2 2s width(W (A)) = inf max{Z ∈ R : KA (Z, − ,− ) = 0}, 2 1+s 1 + s2

−∞ < s < ∞ .

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The order of the polynomial KA (Z, x, y) is n(n − 1). The numerator of the rational function KA (Z, −(1 − s2 )/(1 + s2 ), −(2s)/(1 + s2 )) is given by K0 (Z, s) = (s2 + 1)n(n−1) KA (Z, −

1 − s2 2s ,− ). 2 1+s 1 + s2

The derivative of the rational function K0 (Z, s)/(s2 + 1)n(n−1) with respect to s is given by K0 (Z, s)(s2 + 1) − 2n(n − 1)sK0 (Z, s) . (s2 + 1)n2 −n+1 For a real variable, the condition for K0 (Z, s) = 0 and K0 (Z, s) = 0 is equivalent to K0 (Z, s) = 0 and K0 (Z, s)(s2 + 1) − 2n(n − 1)sK0 (Z, s) = 0. The maximum point or the minimum point of λ1 (H(θ)) − λn (H(θ)), except for θ = π, is attained at some point s0 ∈ R satisfying K0 (Z, s0 ) = 0 and

∂K0 (Z, s) |s=s0 = 0. ∂s

Hence, the values diam(W (A)) and width(W (A)) are the zeros of the resultant of K0 (Z, s) and its derivative with respect to s except for the case θ = π. We give one concrete example to demonstrate the algorithm obtained in Theorem 2.2. The computations were performed in Mathematica. Example 1. Consider the matrix  A=

5 6 2 −3 0 −4

0 8 0

.

Using Mathematica, we obtain the polynomial KA (Z, x, y): KA (Z) = Z 6 − 2(109x2 + 120y 2 )Z 4 + (109x2 + 120y 2 )2 Z 2 −80(2393x6 + 8327x4 y 2 + 785x2 y 4 + 3200y 6 ). We can also obtain the resultant L(Z) of K0 (Z, s) and its derivative K0 (Z, s) with respect to s, which is a constant multiplying the following factor L(Z) = (Z 2 − 160)2 (Z 2 − 40)4 (193Z 2 − 7030)8 (Z 6 − 218Z 4 + 11881Z 2 − 191440)3 ×(759Z 8 − 287243Z 6 + 36014385Z 4 − 1604416800Z 2 + 15634720000)4 . √ The quantity diam(W (A)) is the maximal zero of L(z) which is given by 160  12.6491. The quantity width(W (A)) is given by the maximal zero of the factor Z 6 − 218Z 4 + 1181Z 2 − 191440 which is approximately 12.054. Fig. 1 displays the graph of the curve K(Z, − cos θ, sin θ) = 0 with θ taken as the abscissa.

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Fig. 1. K(Z, − cos θ, − sin θ) = 0.

3. Unitary bordering matrices The n × n Jordan block Jn (0), n ≥ 2, corresponding to the eigenvalue 0 has been discussed in many areas of mathematics. Its numerical range W (Jn (0)) is the circular disc |z| ≤ cos(π/(n + 1)). We may consider Jn (0) as an upper triangular matrix associated to the Dynkin graph of type An . In addition, numerical ranges of upper triangular nilpotent matrices of types Dn and E6 , E7 , E8 are discussed in [2]. The boundary of the numerical range W (Jn (0)) of the matrix Jn (0) is inscribed to the regular (n + 1)-gon with vertices {exp(i(θ +

2kπ )) : k = 0, 1, 2, . . . , n} n+1

with center 0 in the Gaussian plane for any angle 0 ≤ θ < 2π. The regular (n + 1)-gon is inscribed to the unit circle |z| = 1. There is an interesting class of n × n contractions called unitary bordering matrices for which the numerical ranges enjoy this property. More precisely, an n × n matrix A is a unitary bordering matrix, if A is a contraction with dim(Ker(In − A∗ A)) = 1 and absolute values of all eigenvalues of A are strictly less than 1. Assume A is an n × n unitary bordering matrix. The boundary curve of W (A) and the unit circle form a Poncelet pair in the sense that starting from any point on the unit circle, there is an (n + 1)-sided polygon circumscribed to the boundary of W (A) and inscribed to the unit circle |z| = 1. The geometric property of the numerical

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ranges of this class of matrices has been intensively studied in the last few decades (cf. [4,7,8,16,17]). It is shown in [8,17] that the entries of a standard form of an n × n unitary bordering matrix A = (aij ) in the upper triangular form, up to unitary equivalence, are determined by its eigenvalues a1 , a2 , . . . , an in the following way ⎧ ⎪ if i = j ⎪ ⎨a  i j−1 aij = (1 − |ai |2 )(1 − |aj |2 ) if i < j . k=i+1 (−ak ) ⎪ ⎪ ⎩ 0 if i > j For simplicity, we consider an n × n unitary bordering matrix with eigenvalues {a exp(i

2kπ ) : k = 0, 1, 2, . . . , n − 1} n

for 0 < a < 1. If a → 0, this matrix converges to the matrix Jn (0). The standard form (aij ) of this matrix A is given as follows: aij = 0, i > j; akk = a exp(i 2(m−k)π ), k = 1, 2, . . . , n, in the case n = 2m − 1 ≥ 3; n akk = a exp(i 2(m+1−k)π ), k = 1, 2, . . . , n, in the case n = 2m ≥ 2; n aii+1 = 1 − a2 , i = 1, 2, . . . , n − 1; j−1 j−1 aij = (1 − a2 ) k=i+1 (−akk ) = (−1)j−i−1 (1 − a2 ) k=i+1 akk , j ≥ i + 2. The standard form of such an n×n unitary bordering matrix A is denoted by An (a). As a special unitary bordering matrix, the matrix An (a) is unitary similar to the companion matrix of a monic polynomial φ(t) = tn − an . So the matrix An (a) is unitarily similar to the cyclic weighted shift matrix with weights 1, . . . , 1, an . Many interesting results are explored on the numerical ranges of these matrices, such as [6,9,13]. We explicitly formulate diam(W (A)) and width(W (A)) for unitary bordering matrices A of sizes n = 2, 3, 4. Theorem 3.1. Let An (a) be the standard unitary bordering matrix as defined above. (i) For n = 2, diam(W (A2 (a))) = 1 + a2 and width(W (A2 (a))) = 1 − a2 . (ii) For n = 3, diam(W (A3 (a))) = (2 + a6 )1/2 and (A3 (a))) = (2 + (a6 /4))1/2 .  width(W √ 1 (iii) For n = 4, diam(W (A4 (a))) = √2 a8 + 3 + a16 + 2a8 + 8a4 + 5 and  √ width(W (A4 (a))) = √12 a8 + 3 + a16 + 2a8 − 8a4 + 5.

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Proof. For n = 2, the standard form of A2 (a) is given by  A2 (a) =

−a 0

1 − a2 a

 .

Then the numerical range W (A2 (a)) is the elliptical disc: {x + iy : (x, y) ∈ R2 ,

y2 x2 + ≤ 1}, 2 2 (1 + a ) /4 (1 − a2 )2 /4

and thus diam(W (A2 (a))) = 1 + a2 and width(W (A2 (a))) = 1 − a2 . For n = 3, the unitary boarding matrix is given by ⎛

√ a(−1 + 3i)/2 1 − a2 0 a A3 (a) = ⎝ 0 0

⎞ −a(1 − a2 ) ⎠. 1 −√ a2 a(−1 − 3i)/2

Then, we compute the resultant in Theorem 2.2, and obtain that KA (Z, − cos θ, − sin θ) = 32Z 6 − 48(a6 + 2)Z 4 + 18(a6 + 2)2 Z 2 − 2a18 −12a12 + 3a6 − 16 + 27a6 cos(6θ),

(3.1)

0 ≤ θ ≤ π. To simplify the resultant KA (Z, − cos θ, − sin θ), we substitute the variables α = a6 , cos(6θ) = 1 − 2t and Z 2 = z into (3.1), and define   ˜ : α, t) = 32z 3 − 48(α + 2)z 2 + 18(α + 2)2 z − 16 − 3α + 12α2 + 2α3 − 27α(1 − 2t) , K(z 0 < α < 1, 0 ≤ t ≤ 1. Then, the polynomial KA satisfies ˜ : α, t). KA (Z, − cos θ, − sin θ) = KA (Z, − cos θ, − sin θ : a) = K(z ˜ : α, t) has 3 distinct positive zeros in z for 0 < α < 1 and We claim the polynomial K(z 0 ≤ t < 1. Firstly, we show that the function ˜ : α, t) = 16 − 3α + 12α2 + 2α3 − 27α(1 − 2t) −K(0 ˜ : α, 0) = 2(1 − α)2 (8 + α) > 0 and −K ˜ t (0 : α, t) = is strictly positive. In fact −K(0 ˜ 54α > 0 which imply that −K(0 : α, t) > 0 for 0 ≤ t ≤ 1. The derivative ˜ ∂ K(z)/∂z = 96z 2 − 192z − 96αz + 18α2 + 72α + 72

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has two zeros 0 < z1 =

α+2 3(α + 2) < z2 = , 4 4

˜ satisfies and at these points the polynomial K(z) ˜ 1 ) = 54(1 − t)α > 0, K(z except for t = 1, and 1˜ 3 2 3 2 2 − K(z 2 ) = α + 6α + 27tα − 15α + 8 ≥ α + 6α − 15α + 8 = (1 − α) (α + 8) > 0. 2 ˜ : α, t) has 3 distinct positive zeros zj (α, t), j = 1, 2, 3 satisfying It follows that K(z 0 < z1 (α, t) < z1 < z2 (α, t) < z2 < z3 (α, t) ˜ 1 ) = 0 and for 0 ≤ t < 1. The case t = 1 corresponds to cos(6θ) = −1. In this case K(z 6 ˜ z1 = (α+2)/4 is a repeated root of K(z) = 0. We remark that if α = a is fixed, then the ˜ coefficients of the polynomial K(z) are independent of θ except for the constant term −(16 − 3α + 12α2 + 2α3 − 27α(1 − 2t)). ˜ α, t) = 0 is expressed as The equation K(z, 32z 3 − 48(α + 2)z 2 + 18(α + 2)2 z = 16 − 3α + 12α2 + 2α3 − 27α(1 − 2t).

(3.2)

For a fixed α, the polynomial 32z 3 − 48(α + 2)z 2 + 18(α + 2)2 z has a local maximal point 0 < z1 = z1 (α), and a local minimal point z2 = z2 (α) which is greater than z1 (α). Since the right-hand side of (3.2) is increasing in t ∈ [0, 1] and z2 (α) ≤ z3 (α, t), the maximal zero z3 (α, t) increases as t increases on the interval [0, 1]. For t = 1, we have ˜ α, 1) = 2(z − (α + 2))(4z − (α + 2))2 , K(z, and hence z3 (α, 1) = α + 2, z1 (α, 1) = z2 (α, 1) = (α + 2)/4. For t = 0,       ˜ α, 0) = 4z − (α + 8) 8z − (4 + 5α + 3 α(α + 8)) 8z − (4 + 5α − 3 α(α + 8)) . 8K(z, By the inequality,  4 + 5α + 3 α(α + 8) , (α + 8)/4 > 8 the maximal root z3 (α, 0) is given by (α + 8)/4. The conclusion (ii) follows from the determination of z3 (α, 1) and z3 (α, 0).

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For n = 4, the unitary boarding matrix is given by ⎛

a 1 − a2 ⎜0 ia A4 (a) = ⎝ 0 0 0 0

⎞ −ia(1 − a2 ) −ia2 (1 − a2 ) 2 2 a(1 − a ) ⎟ 1−a ⎠. −a 1 − a2 0 −ia

Then, the function KA (Z, − cos θ, − sin θ) becomes   KA (Z, − cos θ, − sin θ) = Z 4 − (a8 + 3)Z 2 + a8 − 2a4 cos(4θ) + 1  2 × 16Z 4 − (8a8 + 24)Z 2 + a16 + 2a8 + 8a4 cos(4θ) + 5 . For a generic θ ∈ [0, 2π], the distinct zeros of the above function in Z are given by 1 ±√ 2

  a8 + 3 ± a16 + 2a8 + 8a4 cos(4θ) + 5,

±

1 2

  a8 + 3 ± 2 a8 − 2a4 cos(4θ) + 1.

We claim that for fixed θ, 1 √ 2



≥

a8 + 3 + 1 2

 a16 + 2a8 + 8a4 cos(4θ) + 5

  a8 + 3 + 2 a8 − 2a4 cos(4θ) + 1.

(3.3)

Since a16 +2a8 −8a4 +5 = (1 −a4 )2 (a8 +2a4 +5) ≥ 0 and a16 +2a8 +8a4 cos(4θ) +5 ≥ 0, we compute the square of the left-hand side of (3.3) minus the square of the right-hand side of (3.3), and obtain that  1  1 8 (a + 3) + ( a16 + 2a8 + 8a4 cos(4θ) + 5 − a8 − 2a4 cos(4θ) + 1). 4 2 Define 

H(a, t) = 2



a16 + 2a8 + 8a4 t + 5 − 2

a8 − 2a4 t + 1 + a8 + 3

for 0 < a ≤ 1, −1 ≤ t ≤ 1. We find the partial derivative of H(a, t) with respect to t: √ Ht (a, t) = 2a4

√ a16 + 2a8 + 8a4 t + 5 + 4 a8 − 2a4 t + 1 √ √ ≥0 a16 + 2a8 + 8a4 t + 5 a8 − 2a4 t + 1

for 0 < a ≤ 1, −1 ≤ t ≤ 1. Therefore, the inequality H(a, t) ≥ H(a, −1) holds. Moreover,  H(a, −1) = (a4 − 1)2 + 2 a16 + 2a8 − 8a4 + 5 ≥ 0.

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Hence, the diameter and the width of W (A4 (a)) are respectively the maximum and the minimum of the function   1 √ a8 + 3 + a16 + 2a8 + 8a4 cos(4θ) + 5, 2 0 ≤ θ ≤ π/2. The assertion (iii) follows. 2 4. Constant width It is obvious that any circle is a curve of constant width. A well known non-circular curve of constant width is given by Franz Reuleaux. The Reuleaux triangle in C is described as {eiθ : 0 ≤ θ ≤ π/3} ∪ {1 + eiθ : 2π/3 ≤ θ ≤ π} √ ∪{(1 + 3i)/2 + eiθ : 4π/3 ≤ θ ≤ 5π/3}. Martini and Mustafaev [14] showed that a curve of constant width can be continuously constructed from a Reuleaux triangle. The Reuleaux triangle has three non-C (1) -smooth √ points. For instance, at the point (1 + 3i)/2, there are two tangents √ (z) −

1 3 1 = ± √ ((z) − ). 2 2 3

If the Reuleaux triangle curve is the boundary of the numerical range of an n × n matrix, then the non-smooth points must be sharp points of the boundary of the numerical range √ (cf. [5, Theorem 5]). In any neighborhood of a sharp point (1 + 3i)/2, the arc of the Reuleaux triangle does not lie on a line. Donoghue’s theorem implies that the boundary of the numerical range at a sharp point z0 is expressed as {z0 + r exp(iθ1 ) : 0 ≤ r1 ≤ } ∪ {z0 + r exp(iθ2 ) : 0 ≤ r ≤ } ( > 0) in a neighborhood of z0 (cf. [10]). Hence, the Reuleaux triangle can not be the boundary of the numerical range of a matrix. This example suggests us a conjecture: If C is a curve of constant width and it is the boundary of the numerical range of a matrix, then C is a circle or a single point. We verify this conjecture for certain class of matrices. For an integer n ≥ 2, denote T (β1 , β2 , . . . , βn−2 , βn−1 ) the n × n nilpotent Toeplitz matrix of the form ⎛

0 β1 ⎜0 0 ⎜ ⎜0 0 ⎜. . ⎜. . ⎜. . ⎝0 0 0 0

β2 β1 0 .. . 0 0

β3 β2 β1 .. . 0 0

. . . βn−2 . . . βn−3 . . . βn−4 .. .. . . ... 0 ... 0

⎞ βn−1 βn−2 ⎟ ⎟ βn−3 ⎟ .. ⎟ ⎟. . ⎟ β1 ⎠ 0

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If n = 2m is even, we define and denote a nilpotent Toeplitz matrix A(β1 , β2 , . . . , βm−1 , βm ) = T (β1 , β2 , . . . , βm−1 , βm , βm−1 , . . . , β2 , β1 ),

(4.1)

while n = 2m − 1 is odd, A(β1 , β2 , . . . , βm−1 ) = T (β1 , β2 , . . . , βm−1 , βm−1 , . . . , β2 , β1 ),

(4.2)

where β1 , . . . , βm−1 are complex numbers and βm is a real number. This type of nilpotent Toeplitz matrices is investigated in [3] for computing the boundary generating curve of the c-numerical range. We give a necessary condition for the boundary of the numerical range of a matrix in this class of matrices to be a curve of constant width. The following result obtained in [3] is crucial for computing the diameter of the numerical range of certain Toeplitz matrices. Theorem 4.1. ([3]) (1) If n = 2m − 1 and B = A(β1 , . . . , βm−1 ) is the n × n Toeplitz matrix (4.2), then the eigenvalues ρk (θ) of the Hermitian matrix HB (θ) are given by  m−1 

ρk (θ) = (−1)k 

βm−j exp(−i(

j=1

(2j − 1)θ (2j − 1)kπ  + )) . n n

(2) If n = 2m and B = A(β1 , . . . , βm ) is the Toeplitz matrix (4.1), then the eigenvalues ρk (θ) of the Hermitian matrix HB (θ) are given by k (βm )

ρk (θ) = (−1)

2

 m−1 

+ (−1)  k

j=1

βm−j exp(−i(

2jθ 2jkπ  + )) , n n

k = 0, 1, 2, . . . , n − 1. The following result determines the numerical range of a matrix of the above nilpotent Toeplitz type for which its boundary is a curve of constant width. Theorem 4.2. Let B be an n × n nilpotent Toeplitz matrix A(β1 , . . . , βm−1 ) (resp. A(β1 , . . . , βm )) for n = 2m − 1 (resp. n = 2m) defined in (4.1) and (4.2). If ∂W (B) is a curve of constant width then W (B) = {0} if n = 2m − 1, and W (B) = {z ∈ C : |z| ≤ a} for some a ≥ 0 if n = 2m. Proof. Assume B is a nilpotent Toeplitz matrix of the form (4.1) or (4.2). Suppose that ∂W (B) is a curve of constant width. Then, the function λ1 (θ) − λn (θ) of HB (θ) is constant for 0 ≤ θ ≤ 2π. Further, there exist 0 ≤ k = ≤ n − 1 such that

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89

λ1 (θ) − λn (θ) = ρk (θ) − ρ (θ) on some interval θ1 ≤ θ ≤ θ2 , thus a constant function on the real line −∞ < θ < ∞ since it is an analytic function. Let βk = ak + ibk , ak , bk ∈ R, and βm = am ∈ R in the case n = 2m is even. To complete the proof of the theorem, we need the following two lemmas. Lemma 4.3. Let n = 2m − 1 be an odd number. If (−1)k+ = 1 and k = , then (−1)k ρk (θ) − (−1)k ρ (θ) = −2

m−1 

am−j sin(

j=1

+2

m−1 

bm−j sin(

j=1

(2j − 1)θ (2j − 1)(k + )π (2j − 1)(k − )π ) sin( + ) 2n n 2n

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) cos( + ). 2n n 2n

(4.3)

If (−1)k+ = −1, then (−1)k ρk (θ) − (−1)k ρ (θ) =2

m−1 

am−j cos(

j=1

+2

m−1 

bm−j cos(

j=1

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) cos( + ) 2n n 2n

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) sin( + ). 2n n 2n

Proof. If (−1)k+ = 1 and k = , then (−1)k ρk (θ) − (−1)k ρ (θ) =

m−1  j=1

+

m−1  j=1

 (2j − 1)θ (2j − 1)kπ (2j − 1)θ (2j − 1) π  am−j cos( + ) − cos( + ) n n n n  (2j − 1)θ (2j − 1)kπ (2j − 1)θ (2j − 1) π  bm−j sin( + ) − sin( + ) n n n n

= −2

m−1 

am−j sin(

j=1

+2

m−1  j=1

bm−j sin(

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) sin( + ) 2n n 2n

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) cos( + ). 2n n 2n

(4.4)

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90

If (−1)k+ = −1, then (−1)k ρk (θ) − (−1)k ρ (θ)  (2j − 1)θ (2j − 1)kπ (2j − 1)θ (2j − 1) π  am−j cos( + ) + cos( + ) n n n n

m−1 

=

j=1

 (2j − 1)θ (2j − 1)kπ (2j − 1)θ (2j − 1) π  + ) + sin( + ) bm−j sin( n n n n

m−1 

+

j=1

=2

m−1 

am−j cos(

j=1

+2

m−1 

bm−j cos(

j=1

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) cos( + ) 2n n 2n

(2j − 1)(k − )π (2j − 1)θ (2j − 1)(k + )π ) sin( + ). 2n n 2n

2

Lemma 4.4. Let n = 2m be an even number. If (−1)k− = 1, then (−1)k ρk (θ) − (−1)k ρ (θ) = −2

m−1 

am−j sin(

j=1

+2

m−1 

bm−j sin(

j=1

j(k − )π 2jθ j(k + )π ) sin( + ) n n n

j(k − )π 2jθ j(k + )π ) cos( + ). n n n

(4.5)

If (−1)k− = −1, then (−1)k ρk (θ) − (−1)k ρ (θ) = am + 2

m−1 

am−j cos(

j=1

+2

m−1 

bm−j cos(

j=1

2jθ j(k + )π j(k − )π ) cos( + ) n n n

j(k − )π 2jθ j(k + )π ) sin( + ). n n n

Proof. If (−1)k− = 1, then  m−1  2jθ 2jkπ  (−1)k ρk (θ) − (−1)k ρ (θ) =  βm−j exp(−i( + )) n n j=1

(4.6)

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91

 m−1  2jθ 2j π  + )) − βm−j exp(−i( n n j=1 =

m−1  j=1

+

 2jθ 2j π  2jθ 2jkπ + ) − cos( + am−j cos( n n n n

m−1  j=1

= −2

 2jθ 2j  2jθ 2jkπ + ) − sin( + ) bm−j sin( n n n n

m−1 

am−j sin(

j=1

+2

m−1 

bm−j sin(

j=1

2jθ j(k + )π j(k − )π ) sin( + ) n n n

2jθ j(k + )π j(k − )π ) cos( + ). n n n

If (−1)k− = −1, then  m−1 

(−1)k ρk (θ) − (−1)k ρ (θ) = am + 

βm−j exp(−i(

j=1

2jθ 2jkπ  + )) n n

 m−1  2jθ 2j π  + + )) βm−j exp(−i( n n j=1 = am + 2

m−1 

am−j cos(

j=1

+2

m−1  j=1

bm−j cos(

2jθ j(k + )π j(k − )π ) cos( + ) n n n

2jθ j(k + )π j(k − )π ) sin( + ). n n n

2

We now proceed to prove Theorem 4.2. We have assumed λ1 (θ) − λn (θ) = ρk (θ) − ρ (θ) for some 0 ≤ k = ≤ n − 1. By Lemmas 4.3 and 4.4, and under the settings (i) n = 2m − 1 is odd, (−1)k+ = 1, (ii) n = 2m − 1 is odd, (−1)k+ = −1, (iii) n = 2m is even, (−1)k+ = 1, the constant assumption ρk (θ) − ρ (θ) implies that ρk (θ) − ρ (θ) vanishes identically. Hence, if one of the above settings holds, the numerical range W (B) reduces to a singleton {z0 }. Obviously, 0 is an eigenvalue of B, and thus 0 ∈ W (B). Hence W (B) = {0}. It remains to deal with the case n = 2m is even and (−1)k+ = −1. By (4.6), the coefficients βj satisfy the condition

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βm−j cos(

j(k − )π ) = 0, n

j = 1, 2, . . . , m − 1. By applying this relation to (4.6), we conclude that ρ0 (θ) − ρ|k−| (θ) = −ρ1 (θ) + ρ1+|k−| (θ) = ρ2 (θ) − ρ2+|k−| (θ) = −ρ3 (θ) + ρ3+|k−| (θ) = · · · are common constant function. By assuming am > 0 and letting p = |k − |, we obtain the equation λ1 (θ) − λn (θ) = ρ0 (θ) − ρp (θ) = ρ2 (θ) − ρ2+p (θ) = ρ4 (θ) − ρ4+p (θ) = · · · = −ρ1 (θ) + ρ1+p (θ) = −ρ3 (θ) + ρ3+p (θ) = · · · , where the index j of ρj (θ) is considered as modulo of n = 2m. Since λ1 (θ) and λn (θ) are respectively the maximal eigenvalue and the minimal eigenvalue of HB (θ), this relation implies that λ1 (θ) = ρ0 (θ) = ρ2 (θ) = ρ4 (θ) = · · · , and λn (θ) = ρ1 (θ) = ρ3 (θ) = ρ5 (θ) = · · · . It follows from (4.6) and the equation ρ0 (θ) − ρ1 (θ) = am that βm−j cos(j

π ) = 0, 2m

j = 1, 2, . . . , m − 1, and hence β1 = β2 = . . . = βm−1 = 0. We conclude that W (B) = {z : |z| ≤ am /2}. 2 The real affine part of the dual curve of the algebraic curve FB (t, x, y) = 0 is called the boundary generating curve of W (B). We follow a similar argument used in [19] and give an expression of the boundary generating curve of the numerical range of some Toeplitz matrices. Theorem 4.5. Let B be the nilpotent Toeplitz matrix A(β, . . . , βm−1 ) or A(β, . . . , βm ) for n = 2m − 1 or n = 2m defined in (4.1) and (4.2). If p(θ) = ρ0 (θ), k = 0 in Theorem 4.1, then the boundary generating curve of W (B) is given by (z(θ)) = p(θ) cos θ − p (θ) sin θ, (z(θ)) = p(θ) sin θ + p (θ) cos θ, 0 ≤ θ ≤ 2π.

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Proof. For n = 2m − 1, define  m−1 

p(θ) = ρ0 (θ) = 

βm−j exp(−i

j=1

(2j − 1)θ  ) , n

0 ≤ θ ≤ 2nπ, and for n = 2m, p(θ) = ρ0 (θ) =

 m−1  am 2jθ  + ) , βm−j exp(−i 2 n j=1

0 ≤ θ ≤ 2nπ. According to the parametric representation method given in [19], the function p(θ) satisfies the equation FB (p(θ), − cos θ, − sin θ) = 0, and hence the affine curve FB (1, x, y) = 0 is parametrized as x=−

cos θ , p(θ)

y=−

sin θ , p(θ)

0 ≤ θ ≤ 2nπ except for the points θ with p(θ) = 0, which gives the conclusion formulate of the dual curve of the algebraic curve FB (t, x, y) = 0. 2 In the above proof, we consider the function ρ0 (θ) for 0 ≤ θ ≤ 2nπ extending the original domain 0 ≤ θ ≤ 2π of the definition. By this extension, the function ρ0 also plays the role of ρj (θ) for j = 1, 2, . . . , n − 1 defined on 0 ≤ θ ≤ 2π. Example 2. Let n = 5, β1 = 6 + 2i, β2 = −1 + i in (4.2). Then the function p(θ) = ρ0 (θ) is given by θ 3θ 3θ θ p(θ) = − cos( ) + sin( ) + 6 cos( ) + 2 sin( ). 5 5 5 5 The equation of the boundary generating curve is given by a polynomial g(x, y) of 30 terms: g(x, y) = 3125(x2 + y 2 )4 − 111250(x2 + y 2 )3 + 500(214x + 899y)(x2 + y 2 )2 +25(31045x2 − 73984xy − 74427y 2 )(x2 + y 2 ) − 4(430384x3 + 1523927x2 y −2460368xy 2 − 1610025y 3 ) + 4(112143x2 − 2096300xy − 3377640y 2 ) +3834616x + 29799000y − 46710280 = 0. Three line segments appear on the boundary of the set W (B). The lines extending these line segments and the other two bitangents of the curve g(x, y) = 0 and one line passing two cusps of the curve define a common polynomial h(x, y) of degree 6 with 28 terms. The boundary generating curve g(x, y) = 0 is displayed in Fig. 2, and the merge the graphics of the two curves g(x, y) = 0 and h(x, y) = 0 is shown in Fig. 3.

94

M.-T. Chien et al. / Linear Algebra and its Applications 582 (2019) 76–98

Fig. 2. Boundary generating curve of W (B).

Fig. 3. Boundary curve and tangent lines.

M.-T. Chien et al. / Linear Algebra and its Applications 582 (2019) 76–98

95

Fig. 4. Boundary generating curve.

Finally, we compare the boundary generating curve of W (A(3, 3/5)) and the curve of constant width by Rabinowitz [19] for the trigonometric polynomial p(θ) =

3261 3 3θ cos2 ( ) + , 25 2 5000

which has the constant width 1.4244. Fig. 4 displays the boundary generating curve of W (A(3, 3/5)), and Fig. 5 is the Rabinowitz curve for the above invariants 3/25, 3261/5000. We merge these two objects in Fig. 6, the image of the boundary generating curve is illustrated in red color and the Rabinowitz curve in blue color. Fig. 7 presents the maximal root of K(Z, θ) = 0 for 0 ≤ θ ≤ π and the constant function Z = 1.4244 which satisfies width(A(3, 3/5)) ∼ 1.41833 < 1.4244 < diam(W (A(3, 3/5)) ∼ 1.43061. Declaration of competing interest The authors declare no conflict of interest. Acknowledgements The authors would like to express their thanks to an anonymous referee for his (or her) valuable suggestions, pointing out an incorrect statement of Theorem 2.2, and also

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Fig. 5. Rabinowitz curve.

Fig. 6. Merging Figs. 5, 6. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

M.-T. Chien et al. / Linear Algebra and its Applications 582 (2019) 76–98

97

Fig. 7. Maximal root of K(Z, θ) = 0.

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