Recall that a

nonsingular matrix B = [[b.sub.i,j]] is called an M-matrix when [b.sub.i,i] > 0 for all i, [b.sub.i,j] [less than or equal to] 0 for alli [not equal to] j, and [B.sup.-1] [greater than or equal to] 0 (elementwise).

Therefore, there exists a unique

nonsingular matrix M [member of] [C.sup.K x ]K such that

This congruence is solvable, since the free term of the matrix polynomial [[parallel][r.sub.uv](x)[parallel].sup.2.sub.1] is a

nonsingular matrix. The unknowns can be found by the method of the indefinite coefficients.

System (6) is normalizable if and only if there exist a

nonsingular matrix P and a matrix Y such that the following LMI

which is a

nonsingular matrix. Since [parallel][E.sub.[epsilon]] - E[parallel][sub.2] = [epsilon], when e is small enough, [E.sub.[espilon]] can approximate E well.

and multiplying a

nonsingular matrix [Q.sub.1] on both sides, we can get

where M [member of] [R.sup.(mf-n)(mf-n)] is a

nonsingular matrix, [[??].sub.y] is the first mf rows of [??], and [[??].sub.u] is the last If rows of [??].

Then R can be characterized as [mathematical expression not reproducible], where [??] [member of] [R.sup.(n-q)x(n-q)] is any

nonsingular matrix.

with [F.sub.11] [member of] [R.sup.n x n], [F.sub.12] [member of] [R.sup.m x n], [F.sub.41] [member of] [R.sup.n x p], [F.sub.42] [member of] [R.sup.m x p], [F.sub.3] [member of] [R.sup.q x p], [G.sub.11] [member of] [R.sup.n x n], [G.sub.12] [member of] [R.sup.m x n], [G.sub.41] [member of] [R.sup.n x p], [G.sub.42] [member of] [R.sup.m x p], [G.sub.3] [member of] [R.sup.q x p], and a

nonsingular matrix M [member of] [R.sup.m x m].

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]).

(1) Let X ~ C[H.sub.m](v, [alpha], [beta], [theta], [OMEGA], kind 1) and let A be an m x m constant

nonsingular matrix. Then, AXA' ~ C[H.sub.m](v, [alpha], [beta], [theta], A[OMEGA]A', kind 1).

If a and b are simultaneously diagonalizable matrix functions, then exists a

nonsingular matrix function v such that both [v.sup.-1]av and [v.sup.-1] bv are diagonal matrix functions.