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

- 49 Let C20) d = max |

Im X. | l^iSn 1 Then from C19) and C20) we obtain n f 2nA—Cn—l)Tr

2A C21> k d

2 £ 5 [lm J S - , i = i where k^n if n is even and k5n-l if n is odd. From C215 we obtain Cl.l). Now, suppose that CI.

2) is true and let = (

R

e \Y~ (

i

m \Y Then we have n n C22) J X

2 = J x

k = Tr

2A - 2A

z £ 0. j =1 k = l From C22) we get that there exists at least one complex number such that Im X ** 0. Let n n ro C23> 5 X

2 = 5 B + 5 = Tr

2A - 2A„ J k k 2 j = 1 k = 1 l=i where B denotes the sum of squares of all real eigenvalues of A and let min Ix I. iSjSm { kjJ I =i C24) x = man fx , x ,...,x 1 = kd i^k

.Sm I k1 2 mJ J Then from C23) and C24) we obtain C25) m x, £ 5 x, 5 Tr A - 2A . k. ^ k. 2 From C25) we get