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''orthogonalization process''
 
''orthogonalization process''
  
An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704201.png" /> an [[Orthogonal system|orthogonal system]] of non-zero vectors generating the same subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704202.png" />. The most well-known is the Schmidt (or Gram–Schmidt) orthogonalization process, in which from a linear independent system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704203.png" />, an orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704204.png" /> is constructed such that every vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704205.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704206.png" />) is linearly expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704207.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704208.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o0704209.png" /> is an upper-triangular matrix. It is possible to construct the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042010.png" /> such that it is orthonormal and such that the diagonal entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042012.png" /> are positive; the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042013.png" /> and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042014.png" /> are defined uniquely by these conditions.
+
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 $.  
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042015.png" />; if the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042016.png" /> have already been constructed, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042017.png" /></td> </tr></table>
+
$$
 +
b _ {i+ 1}  = a _ {i+ 1} + \sum _ { j= 1} ^ { i }  \alpha _ {j} b _ {j} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042018.png" /></td> </tr></table>
+
$$
 
+
\alpha _ {j}  = -  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042019.png" />, are obtained from the condition of orthogonality of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042021.png" />. The geometric sense of this process comprises the fact that at every step, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042022.png" /> is perpendicular to the linear hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042023.png" /> drawn to the end of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042024.png" />. The product of the lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042025.png" /> is equal to the volume of the parallelepiped constructed on the vectors of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042026.png" /> as edges. By normalizing the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042027.png" />, the required orthonormal system is obtained. An explicit expression of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042028.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042029.png" /> is given by the formula
+
\frac{( a _ {j+ 1} , b _ {j} ) }{( b _ {j} , b _ {j} ) }
 
+
,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042030.png" />/''G''<sub>''i''-1</sub></td> </tr></table>
+
$$
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042032.png" /> is the [[Gram determinant|Gram determinant]] of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042033.png" />, with ''G''<sub>0</sub>=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''<sub>''i''</sub>||=SQRT(''G''<sub>''i''</sub>/''G''<sub>''i''-1</sub>). Thus, the corresponding orthonormal system takes the form
+
$  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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042031.png" /> · ''G''<sub>''i''-1</sub> = ''b''<sub>''i''</sub> · SQRT(''G''<sub>''i''-1</sub> / ''G''<sub>''i''</sub>) </td> </tr></table>
+
$$
 +
b _ {i= \left |
  
 +
\begin{array}{cccc}
 +
( a _ {1} , a _ {1} )  &\cdots  &( a _ {1} , a _ {i- 1} )  &a _ {1}  \\
 +
\vdots  &\ddots  &\vdots  & \vdots  \\
 +
( a _ {i} , a _ {1} )  &\cdots  &( a _ {i} , a _ {i- 1} )  &a _ {i}  \\
 +
\end{array}
 +
\right |
 +
$$
 +
(The determinant at the right-hand side has to be formally expanded by the last column). The corresponding orthonormal system takes the form
 +
$$
 +
q _ {i}  = 
 +
\frac{b _ {i} }{\sqrt {G _ {i- 1} G _ {i} } }
 +
,
 +
$$
 +
where  $G_i$ is the [[Gram determinant|Gram determinant]] of the system
 +
$  a _ {1}, \dots, a _ {i} $, with ''G''<sub>0</sub>=1 by definition.
  
 
This process can also be used for a countable system of vectors.
 
This process can also be used for a countable system of vectors.

Latest revision as of 08:33, 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

$$ b _ {i} = \left | \begin{array}{cccc} ( a _ {1} , a _ {1} ) &\cdots &( a _ {1} , a _ {i- 1} ) &a _ {1} \\ \vdots &\ddots &\vdots & \vdots \\ ( a _ {i} , a _ {1} ) &\cdots &( a _ {i} , a _ {i- 1} ) &a _ {i} \\ \end{array} \right | $$ (The determinant at the right-hand side has to be formally expanded by the last column). The corresponding orthonormal system takes the form $$ q _ {i} = \frac{b _ {i} }{\sqrt {G _ {i- 1} G _ {i} } } , $$ where $G_i$ is the Gram determinant of the system $ a _ {1}, \dots, a _ {i} $, with G0=1 by definition.

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=38640
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article