Difference between revisions of "Orthogonalization"
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<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 | <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 | ||
| − | <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" /></td> </tr></table> | + | <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). |
| − | <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" /></td> </tr></table> | + | 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 |
| + | |||
| + | <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> | ||
| − | |||
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 09:48, 25 April 2016
orthogonalization process
An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space 
an orthogonal system of non-zero vectors generating the same subspace in 
. The most well-known is the Schmidt (or Gram–Schmidt) orthogonalization process, in which from a linear independent system 
, an orthogonal system 
is constructed such that every vector 
(
) is linearly expressed in terms of 
, i.e. 
, where 
is an upper-triangular matrix. It is possible to construct the system 
such that it is orthonormal and such that the diagonal entries 
of 
are positive; the system 
and the matrix 
are defined uniquely by these conditions.
The Gram–Schmidt process is as follows. Put 
; if the vectors 
have already been constructed, then
 |
where
 |
, are obtained from the condition of orthogonality of the vector 
to 
. The geometric sense of this process comprises the fact that at every step, the vector 
is perpendicular to the linear hull of 
drawn to the end of the vector 
. The product of the lengths 
is equal to the volume of the parallelepiped constructed on the vectors of the system 
as edges. By normalizing the vectors 
, the required orthonormal system is obtained. An explicit expression of the vectors 
in terms of 
is given by the formula
 /Gi-1 |
where 
is the Gram determinant of the system 
, 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
 · Gi-1 = bi · SQRT(Gi-1 / Gi) |
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) |
Orthogonalization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonalization&oldid=38640