When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram Schmidt process since it is more numerically stable, i . e . rounding errors tend to have less serious effects.
12.
Orthogonalization is also possible with respect to any symmetric bilinear form ( not necessarily an inner product, not necessarily over real numbers ), but standard algorithms may encounter division by zero in this more general setting.
13.
For the definition of a vector space and some further properties I will refer to the article Linear Algebra and Gram-Schmidt Orthogonalization or any textbook in linear algebra and mention only the most important facts for understanding the model.
14.
It is assured that all the vectors in the Gram Schmidt orthogonalization are of length at least 1, and that \ lambda ( L ( B ) ) \ leq \ zeta ( n ) and that 1 \ leq d \ leq \ zeta ( n ) / \ gamma ( n ) where n is the dimension.
15.
An important aspect, with respect to which the following methods differ is whether the orthogonalization of the basis functionals is to be performed over the idealized specification of the input signal ( e . g . gaussian, white noise ) or over the actual realization of the input ( i . e . the pseudo-random, bounded, almost-white version of gaussian white noise, or any other stimulus ).
