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

- 44 Then there exists at least one pair of complex—conjugate eigen-values = ck ± i

dk of A such that Ci. 1) Moreover if CI. 2) then 2nA

2 - Cn—

1>Tr

2A nCn—

1> 2nA_ - Cn-DTr A Tr A £ 2 A, if n is odd if n is even CI. 3) d. £ H 2A

2 - Tr AI if n is odd 2A

2 - Tr A if n is even In the present paper we give another proof of Theorem B. Moreover we prove the following theorems: THEOREM 1. Let A e M O, where H (Z) denote the set of all n * n nxn matrices over Z and let A be non-singular matrix. The necessary and sufficient condition for , j=l,2,...,n to be a roots of unity is |X^ |=1 for j=l

,2, . . . ,n. THEOREM 2. Let A=Ca.

.) be an n*n complex matrix with >- J |det AI

>1 and let = max |A. | where A. are the lSjSn J J characteristic roots of A for j=l,2,...,n, then > 1 + i°s(det A

» . THEOREM 3. Let |X"| = max |A.| where A. for j=l,2,...,n l^jSn J J are characteristic roots of A M CZ>. n£2 and let the n characteristic polynomial of A be irreducible over Z. If for