The optimal precoding and decoding matrices can be chosen from an arbitrary orthogonal matrix
O [member of] [C.sup.MxM] as follows:
X is the orthogonal matrix
composed of the feature vectors corresponding to the eigenvalues.
Every possible rotation R (a 3 x 3 special split orthogonal matrix
) can be constructed from either one of the two related split quaternions q = [q.sub.0]1 + [q.sub.1]i + [q.sub.2]j + [q.sub.3]k or -q = -[q.sub.0]1 - [q.sub.1]i - [q.sub.2]i - [q.sub.3]k using the transformation law :
and [mathematical expression not reproducible] is an orthonormal matrix, [[bar.V].sub.0] [member of] [R.sup.rxr] is an orthogonal matrix
, and [[bar.[summation]].sub.0] [member of] [R.sup.rxr] is a diagonal matrix with nonnegative elements.
it is not difficult to see that Q is just the orthogonal matrix
generated by applying one QR iteration from [B.sup.T]B to [B.sup.T]B with zero shift .
Taking into account the definition of an orthogonal matrix
, the result will also remain an orthogonal matrix
Given a symmetric positive semidefinite matrix W [greater than or equal to] 0, rank (W) [less than or equal to] r, r >:, there exists an orthogonal matrix
U such that
[DELTA](H) [greater than or equal to] 1, and only when H is an orthogonal matrix
, [DELTA](H) = 1.
This technique is based on the theorem from linear algebra which says that a rectangular matrix A can be given as the product of three matrices, as follows: (i) an orthogonal matrix
UA, (ii) a diagonal matrix [summation]A, (iii) the transpose of the orthogonal matrix
Here H represents an orthogonal matrix
. This form normalized to have total mass unity is represented by (dH).
Ultrasound-assisted extraction (UAE) of pectic polysaccharide from oriental tobacco leaves was studied by orthogonal matrix