Numerical range: (in) a matrix nutshell

Transcription

1 Numerical range: (in) a matrix nutshell Panayiotis J. Psarrakos Michael J. Tsatsomeros May 9, 22 Department of Mathematics, National Technical University, Zografou Campus, Athens 578, Greece. Mathematics Department, Washington State University, Pullman, Washington

2 The definition and a name The numerical range, field of values, Wertovorrat, Hausdorff domain, range of values have been, among others, competing names for the same notion. The first two have outlived the competition but the advantage goes decisively to the first: The numerical range of an operator T in a complex Hilbert space H. This is the term coined by Marshall Stone in 932 [33] for the set W (T )={ Tx,x : x H, x =} C. Here T is a bounded linear operator on a separable Hilbert space H with inner product, and induced norm x = x, x. To keep our lives simple, let s confine our discussion to finite dimensions and talk about the numerical range of an n n complex matrix A (A M n (C)), where x = x x is the Euclidean length of x; thatis, W (A) ={x Ax : x C n,x x =}. This is a remarkable set for many reasons. It succinctly captures information about A as a transformation, particularly about the eigenvalues and, perhaps more importantly, about the eigenspaces of A. The shape of W (A) (more accurately its boundary) is also related to the combinatorial structure of A, by which we mean its sign pattern or the pattern of its nonzero entries, sometimes represented by a directed graph. For example, as we shall see in Section 4, an irreducible entrywise nonnegative matrix whose numerical range has 4 maximal elements as depicted on the left of Figure must have cyclic index 4. The numerical range on the right is a circle centered at the origin and thus has an infinite number of maximal elements; the corresponding matrix must also have a special combinatorial structure Figure : Numerical ranges with 4 and with infinite number of maximal elements. Despite the conceptual simplicity of the definition of W (A) (after all it is the image of the Euclidean unit sphere under the continuous map x x Ax ), there are many interesting unanswered questions and fundamental issues surrounding it. Some are curiosity driven and some are driven by its ubiquitous nature in pure and applied mathematics. As a prelude, can you find necessary and sufficient conditions on the matrix A so that W (A)? For the eager reader, more questions like the one above are contained in Section 9. For the enduring reader, we will first review the basic properties of the numerical range of a matrix and focus on results about its convexity and the geometry of its boundary. We will explain how can one draw the boundary of the numerical range and compute the numerical radius, which is the distance from the origin to the point in the numerical range farthest away. We will then be in better position to discuss and appreciate the open problems. Some of them are easy to state and some are of technical nature; all of them though are considered challenging. Some proofs are included but many are omitted or just outlined for expository reasons. The picture we paint of this mathematical area, our bibliography and collection of open problems are by no means comprehensive. They are only meant as a starting place and are certainly biased by our specific interests and point of view. Hilbert, David ( ): The art of doing mathematics consists in finding that special case which contains all the germs of generality. In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 988.

3 2 Basic properties The numerical range of a matrix is known to be a compact and convex subset of C. Thecompactness follows readily from the fact that W (A) is the continuous image of a compact set. The convexity of W (A) is the celebrated Toeplitz-Hausdorff Theorem. In May of 98, Toeplitz [35] showed that the outer boundary of W (A) is a convex curve, and in November of the same year, Hausdorff [7], using a different approach, proved that the numerical range is a convex set. Since then many proofs of the Toeplitz-Hausdorff Theorem have surfaced. In Section 3, we will discuss a proof very close to the original proof of Hausdorff. The spectrum of A, thatis,σ(a) ={λ C :det(λi n A) =}, lies always in W (A). This becomes clear by noticing that when x is a unit eigenvector ( x = ) corresponding to an eigenvalue λ, then λ = x Ax. Thenumerical radius of A is defined by r(a) =max{ z : z W (A)}, which is always greater than or equal to the spectral radius of A, ρ(a) =max{ z : z σ(a)}. Also of interest, though less commonly considered, is the distance between the origin and the boundary W(A), called the inner numerical radius, ˆr(A) =min{ z : z W(A)}. The concepts of numerical range and numerical radius have been studied extensively the last few decades. They are indeed useful tools in studying and understanding matrices and operators (see e.g., [3, 4, 6, 8] and references therein), and have a lot of applications to other subjects such as numerical analysis and differential equations (see e.g., [, 5, 9,, 3, 4, 22, 26, 27, 3, 32] and the discussion in Section 8). From the definition of the numerical range, one can fairly easily deduce the following properties; for details see primarily [8, Chapter ] but also [5]. (P ) For any A M n (C) andforanya, b C, W (aa + bi n )=aw (A)+b. (P 2 ) For any A, B M n (C), W (A + B) W (A)+W (B). (P 3 ) If H(A) =(A+A )/2 ands(a) =(A A )/2 arethehermitian part and the skew-hermitian part of A M n (C), respectively, then Re W (A) =W (H(A)) and Im W (A) =W (S(A)). (P 4 ) For any A M n (C), W (A T )=W(A) andw (A )=W(A) =W (A). As a consequence, if A M n (R), then W (A) is symmetric with respect to the real axis. (A T and A denote the transpose and conjugate transpose of A, respectively; denotes complex conjugation). (P 5 ) For any A M n (C) and any principal submatrix A of A, W (A ) W (A). (Recall that a principal submatrix is obtained by striking out some columns and the corresponding rows.) (P 6 ) For any A M n (C) andb M m (C), W (A B) = convex hull {W (A) W (B)}. (P 7 ) Let X be an n m ( m n) complex matrix such that X X = I m. Then for any A M n (C), W (X AX) W (A). Equality holds if m = n; i.e., W (A) is invariant under unitary similarities. (P 8 ) For any A M n (C), W (A) contains the convex hull of the eigenvalues of A. If A is normal (i.e., A A = AA ), then W (A) equals the convex hull of σ(a). 2

4 (P 9 ) For any A M n (C), W (A) R if and only if A is Hermitian (i.e., A = A); in this case, the endpoints of W (A) coincide with the minimum and the maximum eigenvalues of A. Furthermore, W (A) is a line segment in the complex plane if and only if the matrix A is normal and has collinear eigenvalues; or equivalently, if and only if A = ah + bi n for some a, b C andanhermitianmatrixh. Example In Figure 2, we have sketched first the numerical range of a normal matrix with eigenvalues, ± i, 2 ± i. The numerical range to the right must be of a non-real (complex) matrix as it is not symmetric with respect to the real axis Figure 2: Numerical range of a normal and a non-real matrix. 3 Convexity Most known proofs of the Toeplitz-Hausdorff Theorem (see [7, 8, 8, 24]) begin by observing that the problem reduces to its two-dimensional special case. The original proof of Hausdorff [7] does not use this reduction and is based on the following lemma. Lemma 2 Let H M n (C) be an Hermitian matrix and let µ W (H). Then the set L H (µ) = {x C n : x x =,x Hx = µ} is path connected. Proof By Property (P ) and the second part of Property (P 7 ), without loss of generality, we can assume that µ =andh is a real diagonal matrix, namely, H =diag{d,d 2,...,d n }.Then n n W (H) = d j x j 2 : x,x 2,...,x n C, x j 2 =. j= Suppose now that x =(x j ),y =(y j ) L H (), i.e., x, y are unit vectors such that n j= d j x j 2 = n j= d j y j 2 =. We will show that there is a continuous path in L H () connecting x and y. Since every vector [ r e iθ r 2 e iθ2 r n e iθn ] T L H () (r j, θ j [, 2π) ; j =, 2,...,n) is connected to the real vector [ r r 2 r n ] T L H () by the continuous curve j= [ r e iθ( t) r 2 e iθ2( t) r n e iθn( t) ] T ; t [, ] in L H (), we can assume that x and y are real and entrywise nonnegative. Then the continuous curve ( ) u(t) =(u j (t)) = ( t)x 2 j + ty2 j L H () R n ; t [, ] satisfies u() = x and u() = y, completing the proof. 3

5 Of course, the above lemma holds for every scalar multiple of an Hermitian matrix. Moreover, for a diagonal real matrix H = diag{d,d 2,...,d n } with d d 2 d n, the quantity n j= d j x j 2 (where x,x 2,...,x n C, n j= x j 2 = ) attains its maximum, d,when x = and x 2 = x 3 = = x n = ; it attains its minimum, d n,when x n = and x = x 2 = = x n =. As a consequence, the following holds. Proposition 3 Let H M n (C) be an Hermitian matrix and let µ be either its minimum or its maximum eigenvalue. Then the set L H (µ) ={x C n : x x =, x Hx = µ} comprises all unit eigenvectors of H corresponding to µ. Theorem 4 [Toeplitz-Hausdorff Theorem] For any A M n (C), W (A) is convex. Proof We need to prove that for any distinct points µ, ν W (A), the line segment [µ, ν] lies in W (A). By Property (P ), without loss of generality, we may assume that µ =andν =. Let x, y C n be two unit vectors such that x Ax =andy Ay =, and consider the Hermitian part of A, H(A), and the skew-hermitian part of A, S(A), as in Property (P 3 ). By Lemma 2, the set L S(A) () = {x C n : x x =, x S(A)x =} is path connected. Since x, y L S(A) (), there is a continuous vector function z(t) :[, ] L S(A) () such that z() = x and z() = y. Hence, the function z(t) Az(t) = z(t) H(A)z(t)+z(t) S(A)z(t) = z(t) H(A)z(t) is real and continuous with respect to the variable t, and satisfies z() Az() = x Ax =and z() Az() = y Ay =. Thus,[, ] W (A) and the proof is complete. Remark As it has been proven recently by Lyubich and Markus [25], Lemma 2 is true for every n n complex matrix (n 3). In the special case of 2 2 matrices, the numerical range is always an elliptical disk (with possibly degenerate interior). For a (new) proof of this classical result (stated with details below) see the example in Section 7. Theorem 5 [Elliptical Range Theorem] If A M 2 (C) with eigenvalues λ and λ 2, then the numerical range W (A) is an elliptical disk centered at (/2)trace(A), has foci λ and λ 2, and minor axis length equal to trace(a A) λ 2 λ Boundary and geometry Since the numerical range of every A M n (C) is a compact subset of C, the boundary W(A) and its properties are naturally of special interest. Let s first look at the case where W (A) hasno interior points. Proposition 6 Let A M n (C). Then the following hold: (i) W (A) ={µ} if and only if A = µi n. (ii) W (A) has empty interior (i.e., W (A) = W(A) is a line segment) if and only if trace(a) W(A). n A boundary point µ of a closed region Ω C is said to be a corner of Ω if there is ɛ> such that the intersection of Ω with the circular disk S(µ, ɛ) ={z C : z µ ɛ} is contained in a sector of S(µ, ɛ) of angle less than π. See Figure 3 for an illustration of corners and of the following result. 4

6 Theorem 7 For any A M n (C), ifµ is a corner of W (A), thenµ is an eigenvalue of A. Proof We use a method similar to the one presented in [23]. By Property (P ), without loss of generality, we assume that µ =. Letx C n be a unit vector such that x Ax =,andlety C n be an arbitrary unit vector. Consider the function G y (λ, z) = (x + zy) (λi n A)(x + zy) = (z 2 +2Re(x y)z +)λ ((y Ay)z 2 +2Re(y Ax)z) = (λ λ y (z))r y (z) ; λ, z C, where λ y (z) = (y Ay)z 2 +2Re(y Ax)z z 2 +2Re(x y)z + is analytic in a neighborhood of the origin and satisfies λ y () =, and R y (z) =z 2 +2Re(x y)z + is analytic and does not vanish in a neighborhood of the origin. Clearly, there exists ε> such that for every t (,ε), As a consequence, for every t (,ε), =G y (λ y (t),t)=(x + ty) (λ y (t)i n A)(x + ty). λ y (t) = [(y Ay) t +2Re(y Ax)] t t 2 +2Re(x y) t + W (A), () and since the origin is a corner of W (A), the curve λ y (t) (t (,ε)) lies in a cone K = {ω C : θ Arg ω θ 2, <θ 2 θ ψ <π}. For sufficiently small ε, the ratio t/(t 2 +2Re(x y) t + ) is positive for every t (,ε). Then, by (), the curve λ y (t) (t (,ε)) lies in K for every unit y C n (recall that y is arbitrary) if and only if Ax =. Thus, is an eigenvalue of A and x is a corresponding eigenvector Figure 3: The imaginary numbers i and i (corners) must be eigenvalues. Not every eigenvalue of A on the boundary of W (A) is necessarily a corner of W (A). What we can say, however, is that every eigenvalue µ of A on W(A) is semisimple (i.e., the geometric multiplicity and the algebraic multiplicity of µ are equal), and that every eigenvector of A corresponding to µ is orthogonal to the eigenvectors of A corresponding to the rest of the eigenvalues of A. Theorem 8 Let A M n (C) and let µ be an eigenvalue of A on the boundary W(A). Then A is unitarily similar to a matrix of the form µi m B (m n) such that µ/ σ(b). 5

7 Proof By Schur s Theorem, A is unitarily similar to an upper triangular matrix T whose k first diagonal elements equal to µ, wherek is the algebraic multiplicity of the eigenvalue µ. Ifwe assume that in the first k rows of T there [ is a nonzero ] entry ξ off the main diagonal, then T has µ ξ a principal submatrix of the form T 2 =. By Theorem 5, W (T µ 2 ) is the (nondegenerate) circular disk with center at µ and radius (/2) ξ. By Properties (P 5 )and(p 7 ), it follows that µ IntW (T 2 ) W (A) is an interior point of W (A); a contradiction. Using the results presented so far, one can characterize those complex matrices whose numerical ranges are convex polygons. By the second part of Property (P 8 ), we know that the numerical range of a normal matrix is always a convex polygon. The converse is not always true. Instead, the following holds. Theorem 9 Suppose that the numerical range of A M n (C) is a convex polygon. following hold: Then the (i) If n 4, thena is normal. (ii) If n>4, then either A is normal, or A is unitarily similar to a matrix of the form B C, where B is normal and W (C) W (B). (iii) If W (A) has n or n vertices, then A is normal. Next, we mention some results that capture the essence of what is known about numerical ranges with rotational symmetries. Some more terminology is needed first. The matrix A is called irreducible if there does not exist a permutation matrix P such that [ ] PAP T A, A =,2, A 2,2 where A and A 22 are square, non-vacuous matrices. The matrix A is called k-cyclic if for some permutation matrix P, A, A 2,3.... PAP T = , (2) A k,k A k,... where the zero blocks along the diagonal are square. Notice that a k-cyclic matrix is also m-cyclic for any divisor m of k. The largest positive integer k for which A is k-cyclic is referred to as the cyclic index of A. When A k, =andk 2in(2),A is said to be permutationally similar to a block-shift matrix. The following result, found in [27], is an extension of a result that was originally observed in the unpublished doctoral thesis of J. N. Issos [9] 2. It refers to entrywise nonnegative matrices and parallels an essential part of what is known as the Perron-Frobenius Theorem on the eigenvalues of such matrices. Theorem Let A be a square entrywise nonnegative matrix and assume that H(A) (the Hermitian part of A) is irreducible. Assume also that W (A) is not a circular disk centered at the origin. Then there exists a positive integer k n such that the maximal elements of W (A) are exactly of the form r(a) e i 2tπ k (t =,,...,k ). Infact,k coincides with the cyclic index of A. 2 James Nestor Issos was a student of Emily Haynsworth at Auburn University in the early to mid 6 s. 6

8 This result is illustrated in Figure (left) in the introduction: a matrix satisfying the conditions of Theorem is 4-cyclic if and only if its numerical range has exactly 4 maximal elements. A folk result attributed to J. Anderson states that for any A M n (C) whose numerical range is included in a circular disk D, ifw (A) meets the boundary of D at n+ or more points, then W (A) coincides with D. Based on this result, it is shown in [34] that A is permutationally similar to a block-shift matrix if and only if the numerical range of any matrix with same zero pattern as A is a circular disk centered at the origin. Characterizing matrices whose numerical ranges are circular disks (not necessarily centered at the origin) remains an open problem. 5 Similarity transformations Let A, B M n (C) be two similar matrices, i.e., A = S BS for some invertible S M n (C). If S is unitary (i.e., S = S ), then as we have seen W (A) =W (B). In general, however, there is no tractable relation between W (A) andw (B), except of course that they both contain the convex hull of the common spectrum of A and B (see Property (P 8 )). On the contrary, it is typical that the shape of the numerical range of a matrix changes dramatically under similarity (see Figure 4). In particular, for every µ W (A) \ convex hull {σ(a)}, there exists an invertible matrix S µ such that µ/ W (Sµ AS µ ). This is evident from the following theorem of Givens [2]. Theorem For any A M n (C), convex hull {σ(a)} = dets W (S AS). Remarks (i) When a matrix A M n (C) is diagonalizable, it follows that there is an invertible matrix S M n (C) such that W (S AS) = convex hull {σ(a)}. (ii) The numerical range of a k k Jordan block µ J k (µ) = µ µ is a circular disk centered at µ with radius r(j k ()) = cos (π/(k + )) [36]. Hence, by Property (P 6 ), the numerical range of a matrix J in Jordan Canonical Form coincides with the convex hull of the circular disks that represent the numerical ranges of its Jordan blocks (see Figure 4) Figure 4: Numerical ranges of two Jordan blocks, their direct sum, and a similarity. 7

9 6 Numerical approximation and sketching It is now time to explain how to sketch the numerical range of a matrix as in our figures. The brute force approach would be to plot x Ax for lots and randomly chosen unit vectors x. But that would be too costly, and it would probably not accurately depict the boundary, especially if it is not smooth. Instead, since we know that the numerical range is compact (i.e., closed and bounded), we can try to sketch its boundary (and shade the interior). For that purpose, consider A M n (C) and observe that since W (A) is convex, the boundary of W (A) can be approximated with convex polygons determined by supporting lines of W (A). This approach has indeed a long history [2, 2, 28]) and is based on the following trick: As we have already seen, the numerical range of the Hermitian part of A, H(A) =(A + A )/2, is the closed real interval W (H(A)) = [λ min (H(A)),λ max (H(A))], where λ min (H(A)) and λ max (H(A)) are the minimum and maximum eigenvalues of H(A), respectively. Moreover, by Property (P 3 ), Re W (A) =W (H(A)). Consequently, for any unit x C n, the point µ = x Ax satisfies Re µ =max{reλ : λ W (A)} if and only if Re µ =Re(x Ax) =x H(A)x = λ max (H(A)). Note also that by Proposition 3, x H(A)x = λ max (H(A)) if and only if x is a unit eigenvector of H(A) corresponding to λ max (H(A)). As a consequence, if x is a unit eigenvector of H(A) corresponding to λ max (H(A)), then µ = x Ax is a point farthest to the right of W (A) and so the line {u +iv : u, v R, u= λ max (H(A))} is a supporting line of W (A) at µ. Now, for any θ [, 2π), consider the matrices e iθ A =(cosθ +isinθ)a, H(e iθ A)=cosθH(A) + sin θ (is(a)), the eigenvalue λ θ = λ max (H(e iθ A)), and an associated unit eigenvector x θ ; i.e., H(e iθ A)x θ = λ θ x θ and x θx θ =. Since the line {u +iv : u, v R, u= λ θ } is a supporting line of W (e iθ A)atthepointx θ (eiθ A)x θ, it follows that the line e iθ {u +iv : u, v R, u= λ θ } = {u +iv : u, v R, ucos θ v sin θ = λ θ } is a supporting line of W (A) atx θ Ax θ. Therefore, choosing angles = θ <θ 2 < <θ ρ <θ ρ+ = 2π, we can approximate W (A) (see [8, 2]) by an inscribed convex polygon Q(θ,θ 2,...,θ ρ )with vertices q j = x θ j Ax θj ; j =, 2,...,ρ and by a circumscribed convex polygon P(θ,θ 2,...,θ ρ ) with vertices [ p j = e iθj λ θj + λ ] θ j cos(θ j+ θ j ) λ θj+ ; j =, 2,...,ρ. sin(θ j+ θ j ) Notice that the vertices of P(θ,θ 2,...,θ ρ ) are readily computable since they are the intersection points of the supporting lines of W (A). We can refine the partition of [, 2π] byincreasingρ so that the area difference area (P(θ,θ 2,...,θ ρ )) area (Q(θ,θ 2,...,θ ρ )) = ρ 2 Im ( ) pj p j+ q j q j+ (3) j= (where it is assumed that p ρ+ = p and q ρ+ = q ) is satisfactorily small. This methodology is illustrated in the following algorithm and procedure. 8

10 Algorithm Step Choose = θ <θ 2 < <θ ρ <θ ρ+ =2π. Step 2 For j =, 2,...,ρ, repeat (i) Compute the eigenvalue λ j = λ max (H(e iθj A)). (ii) Compute a unit eigenvector x j = x θj corresponding to λ j. (iii) Compute the j-th vertex of Q(θ,θ 2,...,θ ρ ), q j = x j Ax j. (iv) Compute the j-th vertex of P(θ,θ 2,...,θ ρ ), [ ] λ p j = e iθj θj cos(θ j+ θ j ) λ θj+ λ θj. sin(θ j+ θ j ) Step 3 Set p ρ+ = p and q ρ+ = q, and compute the area difference in (3), E =area(p(θ,θ 2,...,θ ρ )) area (Q(θ,θ 2,...,θ ρ )). Step 4 If E is small enough, then plot P(θ,θ 2,...,θ ρ ). Otherwise, go to Step, and choose a finer partition. An implementation of the above algorithm in MATLAB is available at The above methodology for sketching the boundary can also be formulated in terms of the minimum eigenvalue λ min (H(e iθ A)) instead of λ max (H(e iθ A)), without any essential changes. Moreover, the above discussion yields the following interesting result; see [, 2, 28, 3] for relevant discussion. Theorem 2 For any A M n (C), W(A) is the set of real points of the algebraic curve whose equation in line coordinates is det [ ui n vh(a) w( is(a)) ] =. 7 Point equation One of the biggest challenges surrounding the concept of the numerical range is the unveiling of necessary and/or sufficient conditions for the origin to belong to W (A). More pointedly (no pun intended), one might specifically require the origin to be a boundary or an interior point of the numerical range. The following necessary condition for the boundary case is obtained in [6]. Theorem 3 Let A M n (C) with numerical range W (A), andlet W(A). Then the polynomial f(z) =det(a + za ) satisfies one of the following: (i) Either f(z) is identically zero, i.e., f(z) =for every z C, or (ii) f(z) is a nonzero polynomial of degree at most n with a multiple root of modulus one. Theorem 2 provides the equation of the boundary of the numerical range of a matrix A M n (C) in line coordinates. However, this does not yield the desired point equation for W(A). In contrast, Theorem 3 yields, we believe for first time, the formula of an algebraic curve of degree at most 2n(n ), which contains the boundary W(A). The next result follows directly from Theorem 3 and the simple fact that µ W (A) if and only if W (µi n A). 9

11 Theorem 4 Let A M n (C). Then the boundary of W (A) lies on the algebraic curve {u +iv C :(u, v) R 2, D A (u, v) =}, where D A (u, v) is the discriminant (a.k.a. Sylvester determinant) of the polynomial G A (z; u, v) = det((u +iv)i n A + z [(u +iv)i n A] ) (4) with respect to the variable z. This discriminant is a polynomial (not necessarily with real coefficients) in u, v R of total degree at most 2n(n ). Example 5 Suppose A is a 2 2 complex matrix. By Properties (P )and(p 7 ), for the purpose of studying W (A), we may assume that [ ] a b A =, a where a R and b C. Then the polynomial G A (z; u, v) in (4) is given by The discriminant of G A (z; u, v) is G A (z; u, v) = det((u +iv)i 2 A + z [(u +iv)i 2 A] ) = ((u i v) 2 a 2 ) z 2 +(2u 2 +2v 2 2a 2 b 2 ) z +(u +iv) 2 a 2. D A (u, v) =4 b 2 u 2 +(6a 2 +4 b 2 )v 2 b 4 4a 2 b 2, and thus the algebraic curve D A (u, v) = is the ellipse { W(A) = (u, v) R 2 4u 2 } : 4a 2 + b 2 + 4v2 b 2 =, confirming Theorem 5 (see also Theorem.3.6 in [8]). 8 Numerical radius Consider a matrix A M n (C). The numerical radius r(a) =max{ z : z W (A)} and the inner numerical radius ˆr(A) = min{ z : z W(A)} are of particular interest in many applications. For example, the numerical radius is frequently employed as a more reliable indicator of the rate of convergence of iterative methods than the spectral radius [, 9]. The reason for this is revealed in the following theorem and corollary (for proofs and discussion see [2, 3, 6, 8, 29]). Theorem 6 For any A M n (C), the following hold: (i) (/2) A 2 r(a) A 2, where 2 denotes the spectral (operator) norm induced on M n (C). (ii) For every positive integer m, r(a m ) r(a) m. Corollary 7 For any A M n (C) and for any positive integer m, r(a m ) /m A m /m 2 2 /m r(a m ) /m 2 /m r(a). The numerical radius also plays an important role in the stability analysis of finite difference approximations of solutions to hyperbolic initial value problems [3]. Furthermore, the inner numerical radius has recently been associated with stability issues of Hermitian generalized eigenproblems [5] and of higher order dynamical systems [3]. Using the notation of Section 6, the convexity of W (A) yields directly two basic formulas for the numerical radius and the inner numerical radius of the matrix A.

12 Theorem 8 For any A M n (C), r(a) = max λ max(h(e iθ A)) and ˆr(A) = θ [,2π] min λ max(h(e iθ A)) θ [,2π]. It is now clear that we can compute r(a) andˆr(a) using Step and Step 2 of Algorithm as follows. Algorithm 2 Step Choose = θ <θ 2 < <θ ρ < 2π. Step 2 For j =, 2,...,ρ, compute the eigenvalue λ j = λ max (H(e iθj A)). Step 3 The numerical radius of A is r(a) =max{λ,λ 2,...,λ ρ }, and the inner numerical radius of A is ˆr(A) = min{λ,λ 2,...,λ ρ }. 9 More open problems Problem Let A M n (C), and let s max (A) and s min (A) denote its maximum and minimum singular values (the eigenvalues of A A), respectively. By the first part of Theorem 6, we know that s max (A) = A 2 r(a). Also, if A is Hermitian, then s min (A) ˆr(A) =. Moreover,ifA is normal, then the closest point of W(A) to the origin either coincides with a vertex of W (A) or lies between two vertices of W (A). Thus, it follows readily that s min (A) ˆr(A). So, the following question arises naturally: Is it true that for every A M n (C), s min (A) ˆr(A)? Problem 2 (Originally stated in [6]) Consider a matrix A M n (C) and suppose that the polynomial G A (z; u, v) in (4) is not identically zero and has the following irreducible form G A (z; u, v) = ρ ρ 2 [G,j (u, v)] nj [G 2,j (z,u,v)] mj, j= j= where G,j (u, v)andg 2,j (z,u,v) are mutually distinct (up to scalar multiple) irreducible polynomials in the polynomial ring C[z,u,v], with at least one nonzero power of z in every G 2,j (z,u,v) (j =, 2,...,ρ 2 ), and where the n j s and m j s are positive integers. If m j 2forsomej {, 2,...,ρ 2 }, then the discriminant D A (u, v) of the polynomial G A (z; u, v) (with respect to z) is identically zero providing no information. If we now consider the polynomial ρ G # A (z; u, v) = ρ 2 [G,j (u, v)] nj [G 2,j (z,u,v)], j= then the discriminant D # A (u, v) ofg# A (z; u, v) (with respect to z) is a nonzero polynomial in u, v R, and it seems that the boundary of W (A) lies on the algebraic curve D # A (u, v) =, as in the example bellow. Example 2 Consider the matrix A =. Then the polynomial G A (z; u, v) in (4) is given by j= G A (z; u, v) = (u i v) 4 z 4 +(u i v) 2 (4u 2 +4v 2 3) z 3

13 +6(u 2 + v 2 )(u 2 + v 2 ) z 2 +(u +iv) 2 (4u 2 +4v 2 3) z +(u +iv) 4 = [u +iv +(u iv) z ] 2 [(u i v) 2 z 2 +(2u 2 +2v 2 3) z +(u +iv) 2 ], and its discriminant D A (u, v) (with respect to z ) is identically zero. (u, v) R 2 \ (, ), G A (z; u, v) has a multiple root Note also that for every z(u, v) = u +iv u i v with z(u, v) =. By the above discussion, we consider the (square free) polynomial G # A (z; u, v) =[u +iv +(u i v) z ][(u i v)2 z 2 +(2u 2 +2v 2 3) z +(u +iv) 2 ]. The discriminant of G # A (z; u, v) (with respect to z )is D # A (u, v) = 27(u2 + v 2 )(4u 2 +4v 2 3), and the algebraic curve D # A (u, v) = coincides with the union of the origin and the circle {(u, v) R 2 : u 2 + v 2 =3/4}. One can verify that the numerical range W (A) is the circular disk {u +iv C :(u, v) R 2,u 2 + v 2 3/4}. Problem 3 Beyond the analysis in Section 7 and in the above problem, one of the most important open problems on numerical ranges is the discovery of necessary and/or sufficient conditions for the origin to be a point of W (A). More specifically, it would be interesting to discover conditions for the origin to belong to the boundary or to the topological interior of W (A). Problem 4 What are tractable equivalent conditions so that the numerical range of a matrix is a circular disk? This problem has been discussed at the end of Section 3. It is formally stated for nonnegative matrices (with connected undirected graphs) in [34], where it is solved only for disks centered at the origin. References [] O. Axelsson, H. Lu, and B. Polman, On the numerical radius of matrices and its application to iterative solution methods, Linear and Multilinear Algebra, 37 (994), pp [2] C.A. Berger, On the numerical range of powers of an operator, Notices American Mathematical Society, 2 (965), p. 59. [3] F.F. Bonsall and Duncan, Numerical Ranges I, Cambridge University Press, Cambridge, 97. [4] F.F. Bonsall and Duncan, Numerical Ranges II, Cambridge University Press, Cambridge, 973. [5] S.-H. Cheng and N.J. Higham, The nearest definite pair for the Hermitian generalized eigenvalue problem, Linear Algebra and Its Applications, (999), pp [6] M.-T. Chien, H. Nakazato, and P. Psarrakos, Point equation of the boundary of the numerical range of a matrix polynomial, Linear Algebra and Its Applications, 347 (22), pp [7] C. Davis, The Toeplitz-Hausdorff Theorem explained, Canadian Mathematical Bulletin, 4 (97), pp [8] W. Donoghue, On the numerical range of a bounded operator, Michigan Math. J., 4 (957), pp

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