An

invertible matrix K [member of] M ([Z.sub.N]) is selected.

If [[sigma].sub.l](A) = [[sigma].sub.l](XA[X.sup.-1]) for all A = t[E.sub.12] + s[E.sub.21] [member of] [M.sub.2](C) and A = diag(s, t)[member of] [M.sub.2](C), then there exist [theta] [member of] [0,2[pi]) and an

invertible matrix B [member of][M.sub.2](R) such that X = [e.sup.i[theta]] B.

Hence A is similar to the diagonal matrix diag{[[gamma].sub.1],[[gamma].sub.2], ..., [[gamma].sub.n]] and there exists an

invertible matrix S such that [PHI](t) = S diag{exp([[gamma].sub.1]t), ..., exp([[gamma].sub.n]t)}[S.sup.-1].

If M + nI is an

invertible matrix for every integer n [greater than or equal to] 0, then it can be shown that [GAMMA](M) is also invertible and its inverse corresponds to the inverse of the Gamma function (see [5]).

(a) M is chosen as an

invertible matrix over [Z.sub.q] with eigenvector H([mu]) and eigenvalue 1 so that MA has rank n.

Given an n x n

invertible matrix A, the first-order perturbation of [(A + [DELTA]A).sup.-1] for a small perturbation [DELTA]A is

Since [mathematical expression not reproducible], [there exists]

invertible matrix T(t) [member of] [C.sup.nxn](t), so that

The matrices F(x), G(x) [member of] M(n, C[%]) are called semiscalarly equivalent, if the equality (1) is satisfied for some nonsingular matrix P [member of] M(n, C) and for some

invertible matrix Q(x) [member of] M(n, C[%]) [1] (see also [2]).

Let us consider a family of Fuchsian systems with the fundamental matrix Y(z, a) = [GAMMA](a)[Y.sub.Schl](z, a), where [GAMMA](a) is a holomorphically

invertible matrix. In this case the differential form [omega] = dY(z, a)[Y.sup.-1](z, a) is given by

[r.sub.i] - 1 and an m x m

invertible matrix [GAMMA].

We have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], exploiting the fact that because 5 is invertible modulo 12, U is an

invertible matrix modulo 12.