where [Z.sup.(r)] is the matrix of the labeled set, the matrix [H.sub.n] is a centering matrix, and the matrix [D.sup.(r)] is a

diagonal matrix for the rth media data.

[F.sub.BB][k], which can be got by water-filling is the

diagonal matrix of [P.sub.s].

In our notation, the linear algebraic system [Au.sup.N] = [f.sup.N] is multiplied from the left by the

diagonal matrix[M.sub.C] = [X.sup.H.sub.A] x [X.sub.A] + [X.sup.H.sub.B] x [X.sub.B] is a

diagonal matrix of the fluid-structural coupled modal masses, and [[OMEGA].sub.C] = diag[[[omega].sub.C0], [[omega].sub.C1], [[omega].sub.C2], ...] is a

diagonal matrix of the fluidstructural coupled natural frequencies.

Singular value decomposition (SVD), decomposes a matrix into left and right singular vectors and a

diagonal matrix of singular values.

where [[GAMMA].sub.Rx] is a [N.sub.Rx] x K matrix with its kth column contains [s.sub.Rx] ([[theta].sub.k]), [[GAMMA].sub.Tx] is [N.sub.Tx] x K matrix constructed in a similar manner from [s.sub.Tx]([[theta].sub.k]), [beta] is a K x K

diagonal matrix with its elements containing [[beta].sub.k], and

(i) MIMO systems can be decomposed into a number of single-input single-output (SISO) systems using the unit

diagonal matrix decoupling method.

(i) there is no graph with v [greater than or equal to] 3 (independence of off

diagonal matrix elements);

if [gamma] has multiplicity m> 1 as an eigenvalue of A, and A is

diagonal matrix, we can also choose n independent functions with the form [t.sup.k] exp([gamma]t), k = 0,1, ..., n - 1.

where [mathematical expression not reproducible] is a vector of the disturbance estimate of the observer output and [K.sub.9] [member of] [R.sup.3x3] is a positive definite

diagonal matrix, and [beta] [member of] [R.sup.3] is the intermediate vector for the design.

where [k.sub.1] [member of] [R.sup.3x3] is a

diagonal matrix and each diagonal element of the matrix is a positive constant.

Finally, in the last section, we prove that each symmetric nonsingular idempotent matrix is equivalent to a block

diagonal matrix and a decomposition of the maximal subgroup containing a symmetric idempotent matrix is given (see Theorem 17).