Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1991. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Aleksander Grytczuk and Marek Szal ko ws ki: Spectral proferties of some matrices

- 48 PROOF OF THEOREM B. We have the well-known identity C1Ő) TrCA

2) = Tr

2A - 2A

z . On the other hand we have n 2 n ' C17> TrCA

2) = X

2 .X

2 = 2 [Re \.) - J (lm X.) i =i and CIS) Tr A - ( X

t x

n ) From C16) - C18) we obtain 2 Re X. TrCA

2) = ^ Tr

2A + Cn—

1)Tr A - 2nA. thus •I K - k *•••* kY Cn-1)Tr A - 2nA. n < 0. Since the left hand side cannot be negative for all real X^ then we get that there exists at least ine characteristic root which is a complex number. From C16) - C18) we obtain 2 2 (Re A.)

2- 2 (I- \Y = k i = i L = 1 2 Re X. i = 1 Cn—

1>Tr

2A - 2nA, and therefore we get 2 (i-xj*- 5 »J- K C19) 2 Re X. 2nA -Cn-1)Tr A 2