Difference between revisions of "Orthogonal matrix"

Revision as of 14:52, 7 June 2020

A matrix over a commutative ring with identity for which the transposed matrix coincides with the inverse. The determinant of an orthogonal matrix is equal to . The set of all orthogonal matrices of order over forms a subgroup of the general linear group . For any real orthogonal matrix there is a real orthogonal matrix such that

where

A non-singular complex matrix is similar to a complex orthogonal matrix if and only if its system of elementary divisors possesses the following properties:

1) for , the elementary divisors and are repeated the same number of times;

2) each elementary divisor of the form is repeated an even number of times.

References

[1]	A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)

Comments

The mapping defined by an orthogonal matrix with respect to the standard basis, , , preserves the standard inner product and hence defines an orthogonal mapping. More generally, if and are inner product spaces with inner products , , then a linear mapping such that is called an orthogonal mapping.

Any non-singular (complex or real) matrix admits a polar decomposition with and symmetric and and orthogonal.

References

[a1]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959) pp. 263ff (Translated from Russian)
[a2]	W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. Sect. 43
[a3]	H.W. Turnball, A.C. Aitken, "An introduction to the theory of canonical matrices" , Blackie & Son (1932)

How to Cite This Entry:
Orthogonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_matrix&oldid=48075

This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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-<!--
+A [[Matrix|matrix]] over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703201.png" /> with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703202.png" /> for which the [[Transposed matrix|transposed matrix]] coincides with the inverse. The determinant of an orthogonal matrix is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703203.png" />. The set of all orthogonal matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703204.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703205.png" /> forms a subgroup of the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703206.png" />. For any real orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703207.png" /> there is a real orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o0703208.png" /> such that
-o0703201.png
-$#A+1 = 31 n = 0
-$#C+1 = 31 : ~/encyclopedia/old_files/data/O070/O.0700320 Orthogonal matrix
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
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-{{TEX|auto}}
+<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/o070320/o0703209.png" /></td> </tr></table>
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-A [[Matrix|matrix]] over a commutative ring  $  R $
-with identity  $  1 $
-for which the [[Transposed matrix|transposed matrix]] coincides with the inverse. The determinant of an orthogonal matrix is equal to  $  \pm  1 $.
-The set of all orthogonal matrices of order  $  n $
-over  $  R $
-forms a subgroup of the [[General linear group|general linear group]]  $   \mathop{\rm GL} _ {n} ( R) $.
-For any real orthogonal matrix  $  a $
-there is a real orthogonal matrix  $  c $
-such that
-$$
-cac  ^ {-} 1  =   \mathop{\rm diag}  [\pm  1 \dots \pm  1 , a _ {1} \dots a _ {t} ],
-$$
 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/o070320/o07032010.png" /></td> </tr></table>
-a _ {j}  =  \left \|
-A non-singular complex matrix  $  a $
+A non-singular complex matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032011.png" /> is similar to a complex orthogonal matrix if and only if its system of [[Elementary divisors|elementary divisors]] possesses the following properties:
-is similar to a complex orthogonal matrix if and only if its system of [[Elementary divisors|elementary divisors]] possesses the following properties:
-) for  $  \lambda \neq \pm  1 $,
+) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032012.png" />, the elementary divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032014.png" /> are repeated the same number of times;
-the elementary divisors  $  ( x - \lambda )  ^ {m} $
-and  $  ( x - \lambda  ^ {-} 1 )  ^ {m} $
-are repeated the same number of times;
-) each elementary divisor of the form  $  ( x \pm  1)  ^ {2l} $
+) each elementary divisor of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032015.png" /> is repeated an even number of times.
-is repeated an even number of times.
 ====References====
 <table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Foundations of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR></table>
 ====Comments====
-The mapping  $  \alpha :  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
+The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032016.png" /> defined by an orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032017.png" /> with respect to the standard basis, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032019.png" />, preserves the standard inner product and hence defines an orthogonal mapping. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032021.png" /> are inner product spaces with inner products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032023.png" />, then a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032025.png" /> is called an orthogonal mapping.
-defined by an orthogonal matrix  $  A $
-with respect to the standard basis,  $  \alpha ( x) = Ax $,
-$  x \in \mathbf R  ^ {n} $,
-preserves the standard inner product and hence defines an orthogonal mapping. More generally, if  $  V $
-and  $  W $
-are inner product spaces with inner products  $  \langle   , \rangle _ {V} $,
-$  \langle   , \rangle _ {W} $,
-then a linear mapping  $  \alpha :  V \rightarrow W $
-such that  $  \langle  \alpha ( x) , \alpha ( y) \rangle _ {W} = \langle  x, y \rangle _ {V} $
-is called an orthogonal mapping.
-Any non-singular (complex or real) matrix  $  M $
+Any non-singular (complex or real) matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032026.png" /> admits a [[Polar decomposition|polar decomposition]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032027.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032029.png" /> symmetric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070320/o07032031.png" /> orthogonal.
-admits a [[Polar decomposition|polar decomposition]]  $  M = SQ = Q _ {1} S _ {1} $
-with  $  S $
-and  $  S _ {1} $
-symmetric and  $  Q $
-and  $  Q _ {1} $
-orthogonal.
 ====References====
 <table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1959)  pp. 263ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Noll,   "Finite dimensional spaces" , M. Nijhoff  (1987)  pp. Sect. 43</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.W. Turnball,   A.C. Aitken,   "An introduction to the theory of canonical matrices" , Blackie &amp; Son  (1932)</TD></TR></table>

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Difference between revisions of "Orthogonal matrix"

Revision as of 14:52, 7 June 2020

References

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References