The symbol 0 means the zero matrix
, and the elements of [S.sub.i] (i= 1, 2, 3) are the functions of x, y, z.
(ii) for each [z.sub.[alpha]] [member of] Z there is a fundamental matrix [mathematical expression not reproducible] of (29) such that the matrix [mathematical expression not reproducible] has in the upper right corner the k x(n-k) zero matrix
, and in the lower right corner a (n-k) x (n-k) matrix [[DELTA].sub.[alpha]] with det([[DELTA].sub.[alpha]]) [not equal to] 0.
As illustrated in Figure 1, a well-structured matrix has few nonzero matrix entries outside the blocks (defined as exceptional elements) and few zero matrix
entries inside the blocks (defined as voids).
(vii) -[infinity] denotes the zero matrix
, i.e., the matrix whose entries are all -[infinity].
[0.sub.n x r] [member of] [R.sup.n x r] is the zero rectangular matrix with n rows and r columns, and [I.sub.r], [0.sub.r] [member of] [R.sup.r x r] are the identity matrix and zero matrix
of dimension r, respectively.
Let us define the [S.sub.1] and [S.sub.2] matrices, removing the last row vector and the first row vector, respectively, such that [S.sub.1] = [[I.sub.L-1] [0.sub.1xL-1]] S, [S.sub.2] = [[0.sub.1xL-1] [I.sub.L-1]S, [I.sub.M] M X M Idenotes a M x M identity matrix, and [0.sub.MxN] is a M x N zero matrix
. Sub-matrices [S.sub.1], [S.sub.2] are solved such that:
It is worth mentioning that [He.sub.(0)] can be initialized with the identity matrix and [B.sub.(0)] has to be full-rank matrix chosen differently from the zero matrix
as it is a trivial solution of (20).
where 0 stands for the zero matrix
of dimension (r - 2) x 2, [D.sub.1](z) contains the first two elements of D(z) and [D.sub.2](z) contains the rest of its elements.
where A is state matrix, dim A = n x n, B is the input matrix, dim B = n x r, C is the output matrix, dim C = m x n, D is the zero matrix
, dim D = m x r.
Hypotheses that were established in the LRT for the independence between two groups of variables were [H.sub.0]: [[summation].sub.xy] = [0.sub.p+q] (the two groups of variables are independent) and [H.sub.1]: [[summation].sub.xy] [not equal to] [0.sub.p+q] (the two groups of variables are not independent), where [0.sub.p+q] is the zero matrix
After many trials augmented matrix of size M x N, given by equation (3), formed by combining identity matrix of size M x M, and zero matrix
of size M x (N - M), was found to extract necessary information selectively since it has unity values in the leading diagonal.