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When Is the Numerical Range of a Nilpotent Matrix Circular?

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When Is the Numerical Range of a Nilpotent Matrix Circular? University of Nebraska at Omaha DigitalCommons@UNO Mathematics Faculty Publications Department of Mathematics 3-2010 When is the numerical range of a nilpotent matrix circular? Valentin Matache University of Nebraska at Omaha, [email protected] Mihaela Teodora Matache University of Nebraska at Omaha, [email protected] Follow this and additional works at: https://digitalcommons.unomaha.edu/mathfacpub Part of the Mathematics Commons Recommended Citation Matache, Valentin and Matache, Mihaela Teodora, "When is the numerical range of a nilpotent matrix circular?" (2010). Mathematics Faculty Publications. 24. https://digitalcommons.unomaha.edu/mathfacpub/24 This Article is brought to you for free and open access by the Department of Mathematics at DigitalCommons@UNO. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of DigitalCommons@UNO. For more information, please contact [email protected]. WHEN IS THE NUMERICAL RANGE OF A NILPOTENT MATRIX CIRCULAR? MIHAELA T. MATACHE AND VALENTIN MATACHE Abstract. The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces trM ¤M 2 and trM ¤M 3 are null. The situation becomes more compli- cated as soon as the size is 5. The conditions under which a 5 £ 5 nilpotent matrix has circular numerical range are thoroughly discussed. 2000 Mathematics Subject Classification. Primary 15A60, Secondary 47A12 Keywords. nilpotent matrix, numerical range. 1. Introduction A nilpotent operator is a linear operator T with the property T n = 0 for some positive integer n. We say n is the order of T if n is the least positive integer with the property T n = 0. Recall that the numerical range W (T ) of a bounded operator T acting on some complex Hilbert space is the image of the unit sphere of that space under the quadratic form associated to T , that is W (T ) = fhT x; xi : kxk = 1g. One of the best known properties of numerical ranges is the so called Toeplitz–Hausdorff theorem (see [4, Ch. 22]), saying that numerical ranges are convex subsets of the complex plane. The supremum w(T ) of the absolute values of the complex numbers in W (T ) is called the numerical radius of T . The basic facts of the theory of numerical ranges are contained in [3] and [4]. There are several extensions of the notion of numerical range. Some are inter- esting both in their own right and for the sake of their applications (for instance to quantum information theory, [9]). The numerical ranges of nilpotent operators are insufficiently known as one can deduce from the recent paper [5] where the author uses dilation theory arguments to state and prove that Hilbert space nilpotent op- erators always have circular numerical ranges. That proof is faulty since even 3 £ 3 nilpotent matrices can have non–circular numerical ranges (see [6] for examples of such matrices). Thus it seems both interesting and useful investigating when a n £ n nilpotent matrix with complex entries has circular numerical range. Here are the first remarks on that issue. Remark 1. A 2 £ 2 nilpotent matrix M always has circular numerical range. The disk W (Mp) is centered at the origin and its radius can be calculated with the formula w(M) = tr(M ¤M)=2. Indeed, one of the most popular theorems on numerical ranges, the elliptic range theorem, says that the numerical range of a 2 £ 2 matrix with complex entries is an elliptical disk (possibly degenerate, that is possibly reduced to its focal axis), whose foci are the eigenvalues of that matrix. So, if the matrix is nilpotent, those 1 2 MIHAELA T. MATACHE AND VALENTIN MATACHE eigenvalues equal 0, hence the numerical range is a circular disk centered at the origin (possibly reduced to its center, if we consider the null matrix). The formulas pfor the semi–axes of the elliptical disk (see [3] or [4]), lead to the equality w(M) = tr(M ¤M)=2. The situation when 3 £ 3 nilpotent matrices have circular numerical ranges is completely described in [6, Theorem 4.1], according to which: Remark 2. A 3 £ 3 nilpotent matrix M has circular numerical range if and only if (1) tr(M ¤M 2) = 0; in which case, the numerical radius is computable with the formula p tr(M ¤M) w(M) = : 2 The case of 4 £ 4 matrices is covered by [2]. Indeed, a consequence of [2, Corol- laries 5 and 6] is the following: Remark 3. A 4 £ 4 nilpotent matrix M has circular numerical range if and only if (2) tr(M ¤M 2) = 0 and tr(M ¤M 3) = 0; in which case, the numerical radius is computable with the formula s p tr(M ¤M) + (tr(M ¤M))2 ¡ 64 det(<(M)) w(M) = 8 where <(M) = (1=2)(M + M ¤) denotes the real part of M. In all the cases above, the circular disks are centered at the origin. Hence: Remark 4. A nilpotent k £ k matrix of size 2 · k · 4 has circular numerical range if and only if (3) tr(M ¤M n) = 0; n = 2; 3; 4; 5;::: If the size of a nilpotent matrix M is k, then clearly M n = 0 if n ¸ k, so the only interesting values of n in condition (3) are those between 2 and k. The first reaction is to call (3) the trace condition and try to prove that it is the if and only if characterization of the situation when a nilpotent matrix with complex entries has circular numerical range. As we show in the next section, this is not true, and counterexamples can be produced using 5 £ 5 matrices. After considering such counterexamples, we completely characterize the situation when a nilpotent 5 £ 5 matrix has circular numerical range. This (Theorem 2) is the main result of the current paper. After proving it, we discuss thoroughly when a 5 £ 5 nilpotent matrix satisfying, respectively non–satisfying, the trace condition (3) has circular numerical range. 2. The main results We begin by noting that the trace condition is not necessary for the circularity of the numerical range of a 5 £ 5 matrix. Example 1. The block–diagonal matrix · ¸ B 0 M = 1 0 B2 WHEN IS THE NUMERICAL RANGE OF A NILPOTENT MATRIX CIRCULAR? 3 where 2 3 · ¸ 0 1 1 0 a B = and B = 40 0 15 ; 1 0 0 2 0 0 0 has circular numerical range if jaj is large enough, but tr(M ¤M 2) = 1 6= 0. If jaj is large enough, then W (B2) ⊆ W (B1) since W (B2) is bounded and W (B1) is the circular disk centered at the origin and having radius jaj=2 (by Remark 1). In that case, one has that W (M) = W (B1) because W (M) is the convex hull of the union W (B1) [ W (B2) (a fact valid for any block–diagonal matrix). In the case of 5 £ 5 nilpotent matrices, the trace condition is not sufficient for the circularity of the numerical range. Indeed: Example 2. The matrix 2 3 0 0 1 1 0 6 7 60 0 0 1 17 6 7 (4) M = 60 0 0 0 17 40 0 0 0 05 0 0 0 0 0 satisfies the trace condition, but its numerical range exhibits a flatness on the bound- ary. 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −0.5 0 0.5 1 Figure 1. Numerical Range of M. p It is easy to check that the least eigenvalue of <M, the real part of M, is ¡( 5+ 1)=4. The eigenspace corresponding to it is 2–dimensional and the compression of the imaginary part =M = (1=2i)(M ¡ M ¤) of M to that subspace is easily seen to be a matrix other than a scalar multiple of the identity, thus W (=M) is an interval of the real axis, non–reduced to a point. These facts combinep into showing that the boundary of W (M) has a flatness along the line <z = ¡( 5 + 1)=4, which, of course, establishes the non–circularity of W (M). Recently, the existence of a 4 MIHAELA T. MATACHE AND VALENTIN MATACHE flatness on the boundary of numerical ranges started to be seriously investigated. Reference [1] is an interesting source for such results. Our main goal in this section is to characterize the situation when a 5 £ 5 nilpotent matrix has circular numerical range. To reach it, we need to review first a theorem of R. Kippenhahn, [7]. The reader is also referred to [8] for a recent English translation of reference [7]. Theorem 1 ([8, Theorem 10]). Let M be any square matrix with complex entries. Its numerical range W (M) is the convex hull of the linear envelope of the curve having equation pM (x; y) = 0, where (5) pM (x; y) = det(x<M + y=M + I): Lemma 1. Let M be an n £ n nilpotent matrix with complex entries. If we denote z = x + iy, then equation pM (x; y) = 0 has the following form 0 1 Xk Xk nX¡2l 2l j j 2l (6) 1 + Aljzj + @ (Bljz + Bljz )A jzj = 0 if n = 2k + 1 l=1 l=1 j=1 k = 1; 2; 3;::: respectively 0 1 Xk kX¡1 nX¡2l 2l j j 2l (7) 1 + Aljzj + @ (Bljz + Bljz )A jzj = 0 if n = 2k; l=1 l=1 j=1 k = 2; 3;::: where the coefficients Al are real and Blj complex.
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