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

- 45 some j=l,2,

. . . ,n the root X. is not a root of unity then ixri > 1 + fss^ o where c^2 is some real number and N

q depend only on n and c. We remark that from Theorem 2 it follows immediately: COROLLARY 1. Let A = Ca. .) € M C

2), n 2 2 and det A * ± 1. I J N Then \X\ > 1 + isg-2. . From Corollary 1 it follows that a conjecture of Schinzel and Zassenhaus [23 is true for all matrices A e M (2) with n det A M ± l. PROOF OF THEOREM 1. It is easy to see that if X™ = 1 for j=l

, 2, . . . , n then Suppose that |\.|=1 for j=l,2,...„n and let « CI) fCX) = det CXI-A) be the characteristic polynomial of A. Then we have C2) fCX) = X

n + A X

n_

1 + ... + A 1 n where A. <= Z and A = det A * 0. Since |X. 1=1 for t n J j=l,2,...,n thus by formulas of Viete it follows C3> IV S (Í j for k=l

,2,...,n. From C3) it follows that there exist only finite number of polynomials with integer coefficients such that |=1 for j=l,2, . . . ,n

. Consider the following sequence CI) •• • + A;

m

5X

n_

1 + . . where e 2. The sequence C4) has only finite number of distinct elements, because iVf I = Uj I

m = 1 and IAj m

5 I ^ (jj) ; j,k=l

,2, . . , n.