Namespaces
Variants
Actions

Difference between revisions of "Orthogonal matrix"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (Undo revision 48075 by Ulf Rehmann (talk))
Tag: Undo
Line 1: Line 1:
<!--
+
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.
 
-->
 
  
{{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>
{{TEX|done}}
 
 
 
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 $  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:
 
  
1) for $  \lambda \neq \pm  1 $,  
+
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;
 
  
2) each elementary divisor of the form $  ( x \pm  1)  ^ {2l} $
+
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.
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 $  \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====
 
====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>

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=49346
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article