Page 153 - Textos de Matemática Vol. 44

n−1 βj2 (6.11) c := ̸= 0
and
ON ALMOST NORMAL MATRICES 143
There are various tests known for such polynomials to have all roots on the unit circle, see e.g. [2, 12, 10]. We, however, demonstrate the applicability of Theorem 6.4 in a particular situation when the root location for (6.10) can be handled rather simply.
Corollary 6.5. Let A be an invertible pure almost normal matrix with real eigenvaluesλ1,...,λn−1 correspondingtoitsorthogonaleigenvectors.ThenA lies in the range of the Aluthge transform if and only if
Proof. If all λj are real, they all lie on the same line passing through the origin (namely, the real axis), there is only one cluster, and thus (6.8) takes the form (6.11). On the other hand, (6.10) simplifies to
(6.13) 1(1−ξ)n−2(μξ2 +(c−2Reμ)ξ+μ). μ
It remains to observe that the roots of the quadratic factor in (6.13) lie on the unit circle if and only if its discriminant is non-positive, which is equivalent to (6.12).
If μ is also real, then (6.12) simplifies further to 0 ≤ c ≤ 4.
μ
References
[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations and Operator Theory 13 (1990), no. 3, 307–315.
[2] F. F. Bonsall and M. Marden, Zeros of self-inversive polynomials, Proc. Amer. Math. Soc. 3 (1952), 471–475.
[3] L. Elsner and M. H. C. Paardekooper, On measures of nonnormality of matrices, Linear Algebra Appl. 92 (1987), 107–123.
[4] M. Fiedler, Numerical range of matrices and Levinger’s theorem, Linear Algebra Appl. 220 (1995), 171–180.
[5] K. E. Gustafson and D. K. M. Rao, Numerical range. The field of values of linear operators and matrices, Springer, New York, 1997.
[6] Kh. D. Ikramov, On almost normal matrices, Moscow Univ. Comput. Math. Cybernet. 35 (2011), no. 1, 1–5.
[7] D. Keeler, L. Rodman, and I. Spitkovsky, The numerical range of 3 × 3 matrices, Linear Algebra Appl. 252 (1997), 115–139.
j=1 λj
(6.12) |c − 2 Re μ| ≤ 2 |μ| .