Über Matrizen mit Semidefinitem Realteil, Imaginärteil oder Polarem Defekt und Ihre Eigenwerte

Abstract

We give a new short proof using properties of the field of values to show that

a) a complex matrix with only real eigenvalues is either hermitian or has indefinite imaginary part, and

b) one with only purely imaginary eigenvalues is either skew-hermitian or has indefinite real part, while

c) one whose eigenvalues all have absolute value 1 is either unitary or has indefinite polar defect I—TT^* .

Conversely, every skewsymmetric matrix is the skewsymmetric part of some real matrix that is similar to a real diagonal matrix. The corresponding result for complex matrices is found to be false.

^†New address:Institut fiir Geometrie und Praktische Mathematik der RWTH Aachen, SI Aachen, Templergraben 55.

^†New address:Institut fiir Geometrie und Praktische Mathematik der RWTH Aachen, SI Aachen, Templergraben 55.

Notes

^†New address:Institut fiir Geometrie und Praktische Mathematik der RWTH Aachen, SI Aachen, Templergraben 55.