ELLIPTIC NUMERICAL RANGES OF $4 \times 4$ MATRICES

Trace conditions for symmetry of the numerical range

“…Then there exists a ∈ C such that these eigenvalues are λ 1 = a, λ 2 = ae πi 2 = ai, λ 3 = ae πi = −a, and λ 4 = ae 3πi 2 = −ai. Results in [3] show that if A is nilpotent with the given trace conditions, then W (A) is a disk which clearly has 4-sato. Hence, we may assume a = 0.…”

“…That implication is proved in [11] for 3 × 3 matrices along with other conditions as stated below. We will show that a version of the equivalence of conditions (1) and (2) generalizes to n = 4 but that the n = 4 version of condition (1) no longer implies (3). Tests given in [3] determine exactly when a 4 × 4 matrix M has an elliptic numerical range; a special case of these results is that a nilpotent 4 × 4 matrix has a circular numerical range (which must be centered at the origin due to nilpotence) if and only if tr(M 2 M * ) = 0 and tr(M 3 M * ) = 0.…”

“…We will show that a version of the equivalence of conditions (1) and (2) generalizes to n = 4 but that the n = 4 version of condition (1) no longer implies (3). Tests given in [3] determine exactly when a 4 × 4 matrix M has an elliptic numerical range; a special case of these results is that a nilpotent 4 × 4 matrix has a circular numerical range (which must be centered at the origin due to nilpotence) if and only if tr(M 2 M * ) = 0 and tr(M 3 M * ) = 0. Since circular numerical ranges centered at the origin obviously have n-fold symmetry about the origin for all n, some of the following results for n = 4 can be considered a generalization of the relevant results in [3].…”

Исследование Числовой Области Значений Одной Операторной Матрицы

“…матриц хорошо изучены во многих работах, см. например [11][12][13][14]. В частности, в работе [11] доказано, что ч.о.з.…”

The q-numerical range of 3 by 3 tridiagonal matrices

“…[2], [3], [7]). It seems naturally to ask whether the conditions for elliptical range of W (T ) guarantee that W q (T ) is also elliptical for 0 < q < 1.…”