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

- 4.7 )X\ > 1 + ágidét A j and the proof is complete. PROOF OF THEOREM 3. Let N

q denotes the set of all roots of the characteristic polynomials of fixed degree n £ 2 with integer coefficients and irreducible over Z such that |X| ^ c, where c£2 is some real number. Suppose that for every j=l,2,...,n we have 1 (12) IX.I Sc ° J From (12) we get k . . ÍTfT* C13) |X* I S |X. R ^ c ° <: c for k=l

,2,...,1+N . From (13) and definition of N we get o o that there exists k

c j 5 * k^ '

5 such that k<

j

5,k<

j

5 e 1 2 1 ' 2 e ^1,2, . . . ,l+N

oJ and k t j 3 k Í j 5 (14) 1 = \.

2 for j=l,2,...,n. From (14) we get X'!

1 = 1 for j=l , 2, . . . , n where m = kJ

j

J~k<

jJ and is a root of unity for j=l,2,...,n. By the assumption of our theorem if follows that there exist some j e -il

, 2 , . . . , n such that 1 FTTT (15) |Xj I > c ° From (IS) TTTT TTJT c iri = max IX.I ^ c ° = e ° > 1 + I^J SN J O follows and the proof is complete.