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

- 4.6 Therefore we can take a subsequence -{f CX) r such that CS) f CX) = f CX) = f cx) - ... m_ m _ m O 1 2 where rn < tn < in < . . . . From C4) and CS) we get o i 2 m. m m^ m^ m^ C6) X1 = xaCl)' X2 = X0C2J' ••• ' = ^CTtnJ where oCl), . oCn) denotes some permutation of . There are infinitely many exponents nr such that m

t < m

2 < . . . . On the other hand there exists only n! permutations of the set ,2, . . . ,n^. Therefore there are exponents m. and m. for which m. m, m. ri>. m. m. C7) x

t 1 = x

t 2 , x

2 1 = x

2 2 , ... , x

n « x

n From C7) we get X™ =1 for j=l,2,

...,n where m=m. —m. and m. -m. > 0 and the proof is complete. PROOF OF THEOREM 2. Let fCx)be the characteristic polynomial of the matrix A. It is well-known that C8) IV'

XJ = ,det A

|Since |X

j I 5 I XT I for j=l,2,...,n , thus by (8) it follows C9) \X\ * |det A I 1/

n . Since |det A|>1 thus we have ^ log |det AI A CIO) |det AI = e = 1 + £ log |det A|+ ... From CIO) we get 1 Cll) |det AI

n > 1 + i- log Idet A| . From Cll) and C9) we obtain