Orthogonalization

orthogonalization process

An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space $ V $ an orthogonal system of non-zero vectors generating the same subspace in $ V $. The most well-known is the Schmidt (or Gram–Schmidt) orthogonalization process, in which from a linear independent system $ a _ {1} \dots a _ {k} $, an orthogonal system $ b _ {1} \dots b _ {k} $ is constructed such that every vector $ b _ {i} $ ($ i = 1 \dots k $) is linearly expressed in terms of $ a _ {1} \dots a _ {i} $, i.e. $ b _ {i} = \sum _ {j= 1} ^ {i} \gamma _ {ij} a _ {j} $, where $ C = \| \gamma _ {ij} \| $ is an upper-triangular matrix. It is possible to construct the system $ \{ b _ {i} \} $ such that it is orthonormal and such that the diagonal entries $ \gamma _ {ii} $ of $ C $ are positive; the system $ \{ b _ {i} \} $ and the matrix $ C $ are defined uniquely by these conditions.

The Gram–Schmidt process is as follows. Put $ b _ {1} = a _ {1} $; if the vectors $ b _ {1} \dots b _ {i} $ have already been constructed, then

$$ b _ {i+ 1} = a _ {i+ 1} + \sum _ { j= 1} ^ { i } \alpha _ {j} b _ {j} , $$

where

$$ \alpha _ {j} = - \frac{( a _ {j+ 1} , b _ {j} ) }{( b _ {j} , b _ {j} ) } , $$

$ j = 1 \dots i $, are obtained from the condition of orthogonality of the vector $ b _ {i+ 1} $ to $ b _ {1} \dots b _ {i} $. The geometric sense of this process comprises the fact that at every step, the vector $ b _ {i+ 1} $ is perpendicular to the linear hull of $ a _ {1} \dots a _ {i} $ drawn to the end of the vector $ a _ {i+ 1} $. The product of the lengths $ | b _ {1} | \dots | b _ {k} | $ is equal to the volume of the parallelepiped constructed on the vectors of the system $ \{ a _ {i} \} $ as edges. By normalizing the vectors $ b _ {i} $, the required orthonormal system is obtained. An explicit expression of the vectors $ b _ {i} $ in terms of $ a _ {1} \dots a _ {k} $ is given by the formula

$$ b _ {i} = \left | \begin{array}{llll} ( a _ {1} , a _ {1} ) &\dots &( a _ {1} , a _ {i- 1} ) &a _ {1} \\ \dots &\dots &\dots &{} \\ ( a _ {i} , a _ {1} ) &\dots &( a _ {i} , a _ {i- 1} ) &a _ {i} \\ \end{array} \right | $$

where $$ q _ {i} = \frac{b _ {i} }{\sqrt {G _ {i- 1} G _ {i} } } , $$ is the Gram determinant of the system $ G _ {i} $, with G₀=1 by definition. (The determinant at the right-hand side has to be formally expanded by the last column).

The norm of these orthogonal vectors is given by ||b_i||=SQRT(G_i/G_i-1). Thus, the corresponding orthonormal system takes the form

$ a _ {1} \dots a _ {i} $

This process can also be used for a countable system of vectors.

The Gram–Schmidt process can be interpreted as expansion of a non-singular square matrix in the product of an orthogonal (or unitary, in the case of a Hermitian space) and an upper-triangular matrix with positive diagonal entries, this product being a particular example of an Iwasawa decomposition.

References

[1]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)
[2]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)