Page 88 - Textos de Matemática Vol. 39

78 JUAN-MIGUEL GRACIA
Case 2.2
Let us suppose that |z1 − λk| = |z2 − λk|; see Figure 4. Here σ1 Γk(t) and σ2Γk(t) coincide at t = 0; the first one is increasing and the second one is decreasing. Thus, the same reasoning as in Case 2.1.2 holds and we conclude that the hypotheses of Theorem 2.1 fail. See Figure 3(b).
6 5 4 3 2 1
00 1 2 3 4 5 6
λ2 ...
λk
z2
λ1
z1
Figure 4. Case 2.2.
Examples 2.2. The graphs of Figure 3 correspond to the numbers z1 = 1−1.5 i, z2 = 2−2i and the matrix D = diag(4+ i,1−2i,5) for (a); and numbers z1 = 1 − 1.5 i, z2 = 1 − 2.5 i and matrix D = diag(4 + i, 1 − 2 i, 7) for (b).
Software. All figures were made with Matlab. References
[1] J.M. Gracia, Nearest matrix with two prescribed eigenvalues, Linear Algebra Appl., 401 (2005) 277–294.
[2] Kh.D. Ikramov and A.M. Nazari, Normal matrices and an extension of Malyshev’s for- mula Mathematical Notes, 75 (5) (2004) 608–616.
[3] R.A. Lippert, Fixing two eigenvalues by a minimal perturbation, Linear Algebra Appl., 406 (2005) 177–200.
[4] A.N. Malyshev, A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues, Numer. Math., 83 (3) (1999) 443–454.
Department of Applied Mathematics and Statistics The University of the Basque Country
Faculty of Pharmacy
P.O. Box 450
ES-01080 Vitoria-Gasteiz, Spain
E-mail address: juanmiguel.gracia@ehu.es