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Difference between revisions of "Orthogonalization"

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An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space  $  V $
 
An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space  $  V $
 
an [[Orthogonal system|orthogonal system]] of non-zero vectors generating the same subspace in  $  V $.  
 
an [[Orthogonal system|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} $,  
+
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} $
+
an orthogonal system  $  b _ {1}, \dots, b _ {k} $
is constructed such that every vector  $  b _ {i} $ ($  i = 1 \dots k $)  
+
is constructed such that every vector  $  b _ {i} $ ($  i = 1, \dots, k $)  
is linearly expressed in terms of  $  a _ {1} \dots a _ {i} $,  
+
is linearly expressed in terms of  $  a _ {1}, \dots, a _ {i} $,  
 
i.e.  $  b _ {i} = \sum _ {j= 1}  ^ {i} \gamma _ {ij} a _ {j} $,  
 
i.e.  $  b _ {i} = \sum _ {j= 1}  ^ {i} \gamma _ {ij} a _ {j} $,  
 
where  $  C = \| \gamma _ {ij} \| $
 
where  $  C = \| \gamma _ {ij} \| $
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The Gram–Schmidt process is as follows. Put  $  b _ {1} = a _ {1} $;  
 
The Gram–Schmidt process is as follows. Put  $  b _ {1} = a _ {1} $;  
if the vectors  $  b _ {1} \dots b _ {i} $
+
if the vectors  $  b _ {1}, \dots, b _ {i} $
 
have already been constructed, then
 
have already been constructed, then
  
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$$
 
$$
  
$  j = 1 \dots i $,  
+
$  j = 1, \dots, i $,  
 
are obtained from the condition of orthogonality of the vector  $  b _ {i+ 1} $
 
are obtained from the condition of orthogonality of the vector  $  b _ {i+ 1} $
to  $  b _ {1} \dots b _ {i} $.  
+
to  $  b _ {1}, \dots, b _ {i} $.  
 
The geometric sense of this process comprises the fact that at every step, the vector  $  b _ {i+ 1} $
 
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} $
+
is perpendicular to the linear hull of  $  a _ {1}, \dots, a _ {i} $
 
drawn to the end of the vector  $  a _ {i+ 1} $.  
 
drawn to the end of the vector  $  a _ {i+ 1} $.  
 
The product of the lengths  $  | b _ {1} | \dots | b _ {k} | $
 
The product of the lengths  $  | b _ {1} | \dots | b _ {k} | $
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as edges. By normalizing the vectors  $  b _ {i} $,  
 
as edges. By normalizing the vectors  $  b _ {i} $,  
 
the required orthonormal system is obtained. An explicit expression of 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} $
+
in terms of  $  a _ {1}, \dots, a _ {k} $
 
is given by the formula
 
is given by the formula
  
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The norm of these orthogonal vectors is given by ||''b''<sub>''i''</sub>||=SQRT(''G''<sub>''i''</sub>/''G''<sub>''i''-1</sub>). Thus, the corresponding orthonormal system takes the form
 
The norm of these orthogonal vectors is given by ||''b''<sub>''i''</sub>||=SQRT(''G''<sub>''i''</sub>/''G''<sub>''i''-1</sub>). Thus, the corresponding orthonormal system takes the form
  
$  a _ {1} \dots a _ {i} $
+
$  a _ {1}, \dots, a _ {i} $
  
 
This process can also be used for a countable system of vectors.
 
This process can also be used for a countable system of vectors.

Revision as of 08:23, 13 May 2022


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 ||bi||=SQRT(Gi/Gi-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)
How to Cite This Entry:
Orthogonalization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonalization&oldid=52368
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article