Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

KOSTRA, J. and VAVROS, M., On transformation matrices connected to normal bases in rings

48 J. Kostra, M. Vavros and We have -ft = 50ai, 0

2 . 0

3 = 50«2 , ß

3 . ß

1 = 50a

3 . 0i = -2001 - 1002 , 02 = "2002 - 1003 , 03 — 1001 2003 • and 01 • 02 = -501 + 502 4- 503 , 02 ' 03 — 501 - 502 + 503 , 03 • 01 = 501 + 502 - 503 . And so Z[ai,a-2

, ct

3] is a semiordcr. By the previous examples we have that in the case A = circ

n(ai, ..., a

n)? n «I 7*- ±1, the conjecture from [6], that if a circulant matrix transforms some J=I normal basis of a semiorder to normal basis of its subsemiorder then it transforms any normal basis of any semiorder to normal basis of its subsemiorder, does not hold. Theorem 1. Let A' — circ

n(a, 6,..., 6), a + (n — 1)6= 1. Let A = circ

n(0, b — a,

..., b — a). Let 6=1 (mod n — 1), then matrix A • U, where U is a unimodular circulant matrix of degree n, transforms any norma

1 basis of any semiorder R to a normal basis of its subsemiorder S. Proof. Let A' = circ

n(a, 6,..., 6), a-f (n- 1)6 = 1, A = circ

n(0, 6 — a,

..., 6 — a) and 6=1 (mod n — 1). From we obtain So Then A1 = circ

n ö + (n- 1)6 = 1 6 — a = nb — 1. det A = (-l)

n_

1 • (n - 1) • (nb - l)

n. n- 2 1 (n — 1) • (nb — 1)' (n- 1) • (nb - 1)' * "' (n - I) • (nb - I)