A Note on Product Range of 3-by-3 Normal Matrices

Transcription

1 International Mathematical Forum, Vol. 11, 2016, no. 18, HIKARI Ltd, A Note on Product Range of 3-by-3 Normal Matrices Peng-Ruei Huang Graduate School of Hirosaki University Hirosaki , Japan Hiroshi Nakazato Department of Mathematical Science Faculty of Science and Technology, Hirosaki University Hirosaki , Japan Copyright c 2016 Peng-Ruei Huang and Hiroshi Nakazato. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this note, some invariant under unitary similarity is discussed for 3-by-3 normal matrices. Especially, MATLAB programs for plotting the product ranges are provided. Mathematics Subject Classification: 15A60, 11R29 Keywords: Product range, numerical range, convex set, plotting 1. Introduction Let A be an n n complex matrix. For any integer k with 1 k n, the k-product range is defined to be { k } Wk Π (A) = (UAU ) ii : U is a unitary matrix. i=1

2 886 Peng-Ruei Huang and Hiroshi Nakazato When k equals 1, the product range is the classical numerical range of A W (A) = {x Ax : x C n, x x = 1}. Hence Wk Π (A) is one of a generalized numerical range. Many properties of the classical numerical range and some generalized numerical ranges have been studied and obtained extensively from the last century due to they have been widely applied to many areas, for instance, numerical analysis, perturbation theory and quantum physics, etc [2, 3, 12]. One of the famous results is the Toeplitz-Hausdorff Theorem which indicates that the classical numerical range is always convex for any n n complex matrix A. Indeed, Toeplitz [13] proved the convexity of the boundary W (A) and later Hausdorff [5] proved the simply connectedness of W (A) [4]. Especially, when A is a normal matrix which means A commutes with its conjugate transpose, we have W (A) = Conv(σ(A)) where Conv and σ(a) are the convex hull and the spectrum of A, respectively. In this note, we will concern the convexity of the k-product range of any 3 3 normal matrix. The concept of product ranges was introduced firstly by Marvin Marcus [8] in Bebiano, Li and Providência [1] investigated some geometrical properties, such as convexity, star-shapedness and simply connectedness of Wk Π(A) in In particular, they have shown that W k Π (A) is not generally simply connected when A is an n n normal matrix with n 4. In addition, they also gave an example which shows that W3 Π (A) is not convex when A is a 3 3 normal matrix. Hence Wk Π (A) is not necessary convex in general even when A is normal. However, for the 2-by-2 case, Hu and Tam [6] have shown that W2 Π (A) is a line segment if and only if A is normal. Recently, the authors [7] of this article provide some sufficient conditions for the convexity of W2 Π (A) when A is a 3 3 normal matrix. Before we consider the k-product range of a 3 3 normal matrix, we need the following elementary observation. Let N be an n n normal matrix. As a normal matrix is unitarily diagonalizable, we may assume N is a diagonal matrix whose diagonal entries are the eigenvalues of N, that is, N = diag(λ 1, λ 2,..., λ n ). An n n real matrix A is said to be unistochastic if each entry a ij of A can be expressed as a ij = u ij 2, for some n n unitary matrix U = (u ij ). Note that a unistochastic matrix is a doubly stochatic matrix which means each row and column sums are 1, as U is

3 Product range of 3-by-3 normal matrices 887 a unitary matrix. For any k {1, 2,..., n}, by a simple calculation, we have k (UNU ) ii = i=1 = k n u ij 2 λ j i=1 j=1 k i=1 j=1 n a ij λ j where (a ij ) is a unistochastic matrix. Hence, the k-product range of an n n normal matrix N can be defined equivalently by { k } n Wk Π (N) = a ij λ j : (a ij ) is a unistochastic matrix. i=1 j=1 The equivalent definition is very useful for investigating the product range of any normal matrix. 2. Plotting the product range In this section, we provide two MATLAB programs for plotting the 2- product range and 3-product range of any 3 3 normal matrix. For convenience, they are written as an m-file for MATLAB. The range Wk Π (N) is a compact subset of the Guassian plane C. We approximate it by its finite many representative points. We adopt as the number of representative points in the examples in Section 3. The following program is in order to plot the 2-product range. function y = p(p,q,r,m) a = ones(1,m+1); t = 0:pi/m:pi; s = 0:pi/m:pi; u = 0:pi/m:pi; b1 = kron(a,a); b11 = kron(cos(t),b1); b12 = kron(cos(s),a); b12 = kron(sin(t),b12); b21 = kron(a,sin(u)); b21 = kron(sin(t),b21); b2 = kron(cos(s),sin(u)); bb2 = kron(cos(t),b2); b3 = kron(sin(s),cos(u)); bb3 = kron(a,b3);

4 888 Peng-Ruei Huang and Hiroshi Nakazato b22 = -bb2 - bb3; a11 = b11.^2; a12 = b12.^2; a21 = b21.^2; a22 = b22.^2; a13 = 1 - a11 - a12; a23 = 1 - a21 - a22; a31 = 1 - a11 - a21; a32 = 1 - a12 - a22; a33 = 1 - a31 - a32; X = a11.* p + a12.* q + a13.* r; Y = a21.* p + a22.* q + a23.* r; Z = X.* Y; XX = real(z); YY = imag(z); grid; plot(xx,yy) For plotting the 3-product range of a 3 3 normal matrix, we only need to replace 5 codes from the bottom to the above by following : Z = a31.* p + a32.* q + a33.* r; ZZ = X.* Y.* Z; XX = real(zz); YY = imag(zz); grid; plot(xx,yy) In the above two programs, p, q and r are the diagonal entries of the given normal matrix N. For finer approximation, we can replace m by another large number. Of course, it needs more computational times for larger number m. 3. Examples In this section, we will provide some examples by using the programs in Section 2 and a conjecture will be posed for further work. We set m = 200 in the programs. Example 3.1. Let N = diag(1, ω, ω 2 ) where ω 3 = 1. Figure 1 and Figure 2 are the graphs of W Π 2 (N) and W Π 3 (N), respectively. Note that the diagonal entries of N satisfy Theorem 2.1 in [7], so W Π 2 (N) is a triangle Conv(1, ω, ω 2 ).

5 Product range of 3-by-3 normal matrices 889 Figure 1: W Π 2 (N) Figure 2: W Π 3 (N) Example 3.2. Let N = diag( 75i, 7 21i, 7 21 i). Note that the origin is the inner center of the triangle consisting of the diagonal entries of N. But W2 Π (N) is not convex. See Figure 3. Figure 3: W Π 2 (N) Although the numerical experiments indicate that the product range of a 3 3 normal matrix is not necessary convex, they seem like star-shaped and simply connected. Hence, we end this note by providing the following problem for further researching investigation. Question 3.3. Are the product ranges W Π 2 (N) and W Π 3 (N) for any 3 3 normal matrix always star-shaped or simply connected? Acknowledgements. The second author was supported in part by Japan Society for the Promotion of Science, KAKENHI, project number 15K04890.

6 890 Peng-Ruei Huang and Hiroshi Nakazato References [1] N. Bebiano, C.K. Li and J. da Providência, Product of diagonal elements of matrices, Linear Algebra and its Appl., 178 (1993), [2] N. Bebiano and J. da Providência, Numerical ranges in physics, Linear and Multilinear Algebra, 43 (1998), [3] R. Bhatia, Matrix Analysis, Springer-Verlag, New-York, [4] K.E. Gustafson and D.K.M. Rao, Numerical Range : The Field of Values of Linear Operators and Matrices, Springer-Verlag, New York, [5] F. Hausdorff, Der wertvorrat einer bilinearform, Math. Zeitschrift, 3 (1919), [6] S.A. Hu and T.Y. Tam, On the generalized numerical range with principal character, Linear and Multilinear Algebra, 30 (1991), [7] P.-R. Huang and H. Nakazato, Product of two diagonal entries of a 3-by-3 normal matrix, submitted. [8] M. Marcus, Derivations, Plucker relations, and the numerical range, Indiana Univ. Math. J., 22 (1973), [9] M. Marcus, Matlab and Matrices : A Tutorial, Prentice Hall, [10] H. Nakazato, N. Bebiano and J. da Providência, Product of diagonal entries of the unitary orbit of a 3-by-3 normal matrix, Linear Algebra and its Appl., 429 (2008), [11] H. Nakazato, A. Kovacec, N, Bebiano and J. da Providência, The main diagonal products of 3 3 normal matrices whose eigenvalues are the third roots of unity. The Natalia Bebiano anniversary volume, Textos Mat. Ser. B, 44, Univ. Coimbra, Coimbra, 2013, [12] J. da Providência, The numerical ranges of derivations and quantum physics, Linear and Multilinear Algebra, 37 (1994),

7 Product range of 3-by-3 normal matrices 891 [13] O. Toeplitz, Das algebraische Analogon zu einem Satz von Fejér, Math. Zeitschrift, 2 (1918), Received: July 27, 2016; Published: September 19, 2016