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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
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<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
 
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 <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:
+
\begin{array}{rc}
 +
\cos  \phi _ {j}  &\sin  \phi _ {j}  \\
 +
- \sin  \phi _ {j}  &\cos  \phi _ {j}  \\
 +
\end{array}
 +
\right \| .
 +
$$
  
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;
+
A non-singular complex matrix  $  a $
 +
is similar to a complex orthogonal matrix if and only if its system of [[Elementary divisors|elementary divisors]] possesses the following properties:
  
2) 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.
+
1) for  $  \lambda \neq \pm  1 $,
 +
the elementary divisors  $  ( x - \lambda )  ^ {m} $
 +
and  $  ( x - \lambda  ^ {-} 1 )  ^ {m} $
 +
are repeated the same number of times;
 +
 
 +
2) each elementary divisor of the form $  ( x \pm  1)  ^ {2l} $
 +
is repeated an even number of times.
  
 
====References====
 
====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>
 
<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====
 
====Comments====
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.
+
The mapping $  \alpha : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
 +
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 <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.
+
Any non-singular (complex or real) matrix $  M $
 +
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====
 
====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>
 
<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>

Latest revision as of 14:54, 7 June 2020


A matrix over a commutative ring $ R $ with identity $ 1 $ for which the 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 $ \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

$$ a _ {j} = \left \| \begin{array}{rc} \cos \phi _ {j} &\sin \phi _ {j} \\ - \sin \phi _ {j} &\cos \phi _ {j} \\ \end{array} \right \| . $$

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

1) for $ \lambda \neq \pm 1 $, the elementary divisors $ ( x - \lambda ) ^ {m} $ and $ ( x - \lambda ^ {-} 1 ) ^ {m} $ are repeated the same number of times;

2) each elementary divisor of the form $ ( x \pm 1) ^ {2l} $ 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 $ \alpha : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ 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 $ admits a polar decomposition $ M = SQ = Q _ {1} S _ {1} $ with $ S $ and $ S _ {1} $ symmetric and $ Q $ and $ Q _ {1} $ 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=17418
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article