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''for quadratic forms''
 
''for quadratic forms''
  
 
The theorem stating that for any way of reducing a quadratic form (cf. also [[Quadratic forms, reduction of|Quadratic forms, reduction of]])
 
The theorem stating that for any way of reducing a quadratic form (cf. also [[Quadratic forms, reduction of|Quadratic forms, reduction of]])
  
<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/l/l057/l057710/l0577101.png" /></td> </tr></table>
+
$$
 +
\sum _ { i,j= } 1 ^ { s }
 +
a _ {ij} x _ {i} x _ {j}  $$
  
 
with real coefficients to a sum of squares
 
with real coefficients to a sum of squares
  
<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/l/l057/l057710/l0577102.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 1 ^ { s }
 +
b _ {i} y _ {i}  ^ {2}
 +
$$
  
 
by a linear change of variables
 
by a linear change of variables
  
<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/l/l057/l057710/l0577103.png" /></td> </tr></table>
+
$$
 +
( x _ {1} \dots x _ {s} )  = \
 +
( y _ {1} \dots y _ {s} ) Q ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577104.png" /> is a non-singular matrix with real coefficients, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577105.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577106.png" />) of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577107.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577108.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l0577109.png" />) is fixed. In its classical form, the law of inertia was established by J.J. Sylvester. This statement is sometimes called Sylvester's theorem.
+
where $  Q $
 +
is a non-singular matrix with real coefficients, the number $  p $(
 +
respectively, $  n $)  
 +
of indices $  i $
 +
for which $  b _ {i} > 0 $(
 +
or $  b _ {i} < 0 $)  
 +
is fixed. In its classical form, the law of inertia was established by J.J. Sylvester. This statement is sometimes called Sylvester's theorem.
  
In its modern form, the law of inertia is the following statement concerning properties of symmetric bilinear forms over ordered fields. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771010.png" /> be a finite-dimensional vector space over an ordered field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771011.png" />, endowed with a non-degenerate symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771012.png" />. Then there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771013.png" /> such that for any orthogonal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771015.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771016.png" /> there exist among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771017.png" /> elements
+
In its modern form, the law of inertia is the following statement concerning properties of symmetric bilinear forms over ordered fields. Let $  E $
 +
be a finite-dimensional vector space over an ordered field $  k $,  
 +
endowed with a non-degenerate symmetric bilinear form $  f $.  
 +
Then there exists an integer $  p \geq  0 $
 +
such that for any orthogonal basis $  e _ {1} \dots e _ {s} $
 +
in $  E $
 +
with respect to $  f $
 +
there exist among the $  s $
 +
elements
  
<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/l/l057/l057710/l05771018.png" /></td> </tr></table>
+
$$
 +
f ( e _ {i} , e _ {i} ) ,\ \
 +
i = 1 \dots s ,
 +
$$
  
exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771019.png" /> positive and exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771020.png" /> negative ones. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771021.png" /> is called the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771022.png" />, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771023.png" /> its index of inertia. Two equivalent forms have the same signature. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771024.png" /> is a [[Euclidean field|Euclidean field]], equality of signatures is a sufficient condition for the equivalence of bilinear forms. If the index of inertia <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771025.png" />, the form is called positive definite, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771026.png" />, negative definite. These cases are characterized by the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771027.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771028.png" />) for any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771029.png" />. It follows from the law of inertia that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771030.png" /> is an orthogonal direct sum (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771031.png" />) of subspaces,
+
exactly $  p $
 +
positive and exactly $  n = s - p $
 +
negative ones. The pair $  ( p , n) $
 +
is called the signature of $  f $,  
 +
and the number $  n $
 +
its index of inertia. Two equivalent forms have the same signature. If $  k $
 +
is a [[Euclidean field|Euclidean field]], equality of signatures is a sufficient condition for the equivalence of bilinear forms. If the index of inertia $  n = 0 $,  
 +
the form is called positive definite, and for $  p = 0 $,  
 +
negative definite. These cases are characterized by the property that $  f( x, x) > 0 $(
 +
respectively, $  f ( x , x ) < 0 $)  
 +
for any non-zero $  x \in E $.  
 +
It follows from the law of inertia that $  E $
 +
is an orthogonal direct sum (with respect to $  f  $)  
 +
of subspaces,
  
<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/l/l057/l057710/l05771032.png" /></td> </tr></table>
+
$$
 +
= E _ {+} \oplus E _ {-} ,
 +
$$
  
such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771034.png" /> is positive definite while the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771036.png" /> is negative definite and
+
such that the restriction of $  f $
 +
to $  E _ {+} $
 +
is positive definite while the restriction of $  f $
 +
to $  E _ {-} $
 +
is negative definite and
  
<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/l/l057/l057710/l05771037.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  E _ {+}  = p ,\  \mathop{\rm dim}  E _ {-= n
 +
$$
  
(so that the dimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771039.png" /> do not depend on the decomposition).
+
(so that the dimensions of $  E _ {+} $
 +
and $  E _ {-} $
 +
do not depend on the decomposition).
  
Sometimes the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771040.png" /> is taken to be the difference
+
Sometimes the signature of $  f $
 +
is taken to be the difference
  
<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/l/l057/l057710/l05771041.png" /></td> </tr></table>
+
$$
 +
\sigma ( f  )  = p - n .
 +
$$
  
If two forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771043.png" /> determine the same element of the [[Witt ring|Witt ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771044.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771046.png" />. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771048.png" /> for any non-degenerate forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771051.png" />, so that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771052.png" /> defines a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771053.png" /> into the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771054.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771055.png" /> is a Euclidean field, then this homomorphism is an isomorphism.
+
If two forms $  f $
 +
and $  g $
 +
determine the same element of the [[Witt ring|Witt ring]] $  W ( k) $
 +
of the field $  k $,  
 +
then $  \sigma ( f  ) = \sigma ( g) $.  
 +
Furthermore, $  \sigma ( f _ {1} \oplus f _ {2} ) = \sigma ( f _ {1} ) + \sigma ( f _ {2} ) $
 +
and $  \sigma ( f _ {1} \otimes f _ {2} ) = \sigma ( f _ {1} ) \sigma ( f _ {2} ) $
 +
for any non-degenerate forms $  f _ {1} $
 +
and $  f _ {2} $,  
 +
and $  \sigma ( \langle  1 \rangle ) = 1 $,  
 +
so that the mapping $  f \rightarrow \sigma ( f  ) $
 +
defines a homomorphism from $  W ( k) $
 +
into the ring of integers $  \mathbf Z $.  
 +
If $  k $
 +
is a Euclidean field, then this homomorphism is an isomorphism.
  
The law of inertia can be generalized to the case of a Hermitian bilinear form over a maximal ordered field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771056.png" />, over a quadratic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771057.png" /> or over the skew-field of quaternions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771058.png" /> (see [[#References|[1]]], [[#References|[4]]]).
+
The law of inertia can be generalized to the case of a Hermitian bilinear form over a maximal ordered field $  k $,  
 +
over a quadratic extension of $  k $
 +
or over the skew-field of quaternions over $  k $(
 +
see [[#References|[1]]], [[#References|[4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2272929}} {{MR|0928386}} {{MR|0896478}} {{MR|0782297}} {{MR|0782296}} {{MR|0722608}} {{MR|0682756}} {{MR|0643362}} {{MR|0647314}} {{MR|0610795}} {{MR|0583191}} {{MR|0354207}} {{MR|0360549}} {{MR|0237342}} {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.02101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) {{MR|0506372}} {{ZBL|0292.10016}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2272929}} {{MR|0928386}} {{MR|0896478}} {{MR|0782297}} {{MR|0782296}} {{MR|0722608}} {{MR|0682756}} {{MR|0643362}} {{MR|0647314}} {{MR|0610795}} {{MR|0583191}} {{MR|0354207}} {{MR|0360549}} {{MR|0237342}} {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.02101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) {{MR|0506372}} {{ZBL|0292.10016}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The name index is also used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057710/l05771059.png" />.
+
The name index is also used for $  \min ( p, n) $.

Revision as of 22:16, 5 June 2020


for quadratic forms

The theorem stating that for any way of reducing a quadratic form (cf. also Quadratic forms, reduction of)

$$ \sum _ { i,j= } 1 ^ { s } a _ {ij} x _ {i} x _ {j} $$

with real coefficients to a sum of squares

$$ \sum _ { i= } 1 ^ { s } b _ {i} y _ {i} ^ {2} $$

by a linear change of variables

$$ ( x _ {1} \dots x _ {s} ) = \ ( y _ {1} \dots y _ {s} ) Q , $$

where $ Q $ is a non-singular matrix with real coefficients, the number $ p $( respectively, $ n $) of indices $ i $ for which $ b _ {i} > 0 $( or $ b _ {i} < 0 $) is fixed. In its classical form, the law of inertia was established by J.J. Sylvester. This statement is sometimes called Sylvester's theorem.

In its modern form, the law of inertia is the following statement concerning properties of symmetric bilinear forms over ordered fields. Let $ E $ be a finite-dimensional vector space over an ordered field $ k $, endowed with a non-degenerate symmetric bilinear form $ f $. Then there exists an integer $ p \geq 0 $ such that for any orthogonal basis $ e _ {1} \dots e _ {s} $ in $ E $ with respect to $ f $ there exist among the $ s $ elements

$$ f ( e _ {i} , e _ {i} ) ,\ \ i = 1 \dots s , $$

exactly $ p $ positive and exactly $ n = s - p $ negative ones. The pair $ ( p , n) $ is called the signature of $ f $, and the number $ n $ its index of inertia. Two equivalent forms have the same signature. If $ k $ is a Euclidean field, equality of signatures is a sufficient condition for the equivalence of bilinear forms. If the index of inertia $ n = 0 $, the form is called positive definite, and for $ p = 0 $, negative definite. These cases are characterized by the property that $ f( x, x) > 0 $( respectively, $ f ( x , x ) < 0 $) for any non-zero $ x \in E $. It follows from the law of inertia that $ E $ is an orthogonal direct sum (with respect to $ f $) of subspaces,

$$ E = E _ {+} \oplus E _ {-} , $$

such that the restriction of $ f $ to $ E _ {+} $ is positive definite while the restriction of $ f $ to $ E _ {-} $ is negative definite and

$$ \mathop{\rm dim} E _ {+} = p ,\ \mathop{\rm dim} E _ {-} = n $$

(so that the dimensions of $ E _ {+} $ and $ E _ {-} $ do not depend on the decomposition).

Sometimes the signature of $ f $ is taken to be the difference

$$ \sigma ( f ) = p - n . $$

If two forms $ f $ and $ g $ determine the same element of the Witt ring $ W ( k) $ of the field $ k $, then $ \sigma ( f ) = \sigma ( g) $. Furthermore, $ \sigma ( f _ {1} \oplus f _ {2} ) = \sigma ( f _ {1} ) + \sigma ( f _ {2} ) $ and $ \sigma ( f _ {1} \otimes f _ {2} ) = \sigma ( f _ {1} ) \sigma ( f _ {2} ) $ for any non-degenerate forms $ f _ {1} $ and $ f _ {2} $, and $ \sigma ( \langle 1 \rangle ) = 1 $, so that the mapping $ f \rightarrow \sigma ( f ) $ defines a homomorphism from $ W ( k) $ into the ring of integers $ \mathbf Z $. If $ k $ is a Euclidean field, then this homomorphism is an isomorphism.

The law of inertia can be generalized to the case of a Hermitian bilinear form over a maximal ordered field $ k $, over a quadratic extension of $ k $ or over the skew-field of quaternions over $ k $( see [1], [4]).

References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[2] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3] E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[4] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) MR0506372 Zbl 0292.10016

Comments

The name index is also used for $ \min ( p, n) $.

How to Cite This Entry:
Law of inertia. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Law_of_inertia&oldid=47594
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article