Section III describes stochastic processor of the orthogonal transformation
. Section IV exposes theory of operation.
Only with orthogonal transformation
, the original vectors can be invariant to their Euclidean distance or angle with each other.
Let [Q.sup.(j+1).sub.i] [member of] [C.sup.(j+1)Lx(j+1)L] be the orthogonal transformation
which annihilates all subdiagonal entries in columns (i - 1)L +1 to iL of [[bar.H].sup.(B).sub.j] and effects no other rows so that we can write
In particular, it uses an orthogonal transformation
to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.
PCA is a statistical procedure that uses orthogonal transformation
to reveal the internal structure of data, which is CME (Jackson, 1991; Williams et al., 2004).
The Varimax procedure finds the orthogonal transformation
of the loading matrix that maximizes the sum of those variances, summing across all m rotated factors, while the Quartimax procedure performs the same transformation across all p variables.
Number of techniques were developed for enforcing EBCs in the problem like Lagrange multiplier, penalty method, orthogonal transformation
techniques, coupling with FEM, Nitsche's method, singular weighing functions, boundary collocation, and D'Alembert's Principle.
In order to improve the numerical instability associated with the recursive least squares(RLS) algorithm, a desirable technique based on numerically stable, robust orthogonal triangularization approach that computes the information matrix directly via QR decomposition is considered [1,3,5,6,9].The triangularization process can be realized by any of the orthogonal transformation
components; Givens rotations, Householder transformation and Modified Gram-Schmidt orthogonalization [6,8].
We define the orthogonal transformation
matrix Q(n) as in (2.9) and (2.10) as a partition matrix
An orthogonal transformation
is used to annihilate it which produces a fill-in in `b', whose annihilation produces `c'.
When uncertainties are statistically correlated, Rosenblatt transformation, Nataf transformation, and orthogonal transformation
are often used to handle these uncertainties in reliability analysis.
The unstructured orthogonal transformation
, however, shows the expected problems with purely imaginary eigenvalues.