Namespaces
Variants
Actions

Difference between revisions of "Gram determinant"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
g0447401.png
 +
$#A+1 = 33 n = 0
 +
$#C+1 = 33 : ~/encyclopedia/old_files/data/G044/G.0404740 Gram determinant
 +
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}}
 +
{{TEX|done}}
 +
 
A determinant of the form
 
A determinant of 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/g/g044/g044740/g0447401.png" /></td> </tr></table>
+
$$
 +
\Gamma  = \Gamma ( a _ {1} \dots a _ {n} )  = \
 +
\mathop{\rm det}  | ( a _ {i} , a _ {k} ) | ,\ \
 +
i, k = 1 \dots n,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447402.png" /> are elements of a (pre-)Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447403.png" /> are their scalar products. A Gram determinant is equal to the square of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447404.png" />-dimensional volume of the [[Parallelotope|parallelotope]] constructed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447405.png" />.
+
where $  a _ {1} \dots a _ {n} $
 +
are elements of a (pre-)Hilbert space and $  ( a _ {i} , a _ {k} ) $
 +
are their scalar products. A Gram determinant is equal to the square of the $  n $-
 +
dimensional volume of the [[Parallelotope|parallelotope]] constructed on $  a _ {1} \dots a _ {n} $.
  
 
A Gram determinant is the determinant of a non-negative Hermitian form
 
A Gram determinant is the determinant of a non-negative Hermitian 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/g/g044/g044740/g0447406.png" /></td> </tr></table>
+
$$
 +
\sum _ {i, k = 1 } ^ { n }
 +
( a _ {i} , a _ {k} )
 +
\xi _ {i} \overline{ {\xi _ {k} }}\; = \
 +
\left \|
 +
\sum _ {i = 1 } ^ { n }  a _ {i} \xi _ {i} \
 +
\right \|  ^ {2} ,
 +
$$
  
 
which determines its basic properties:
 
which determines its basic properties:
  
1) A Gram determinant is non-negative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447407.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g0447408.png" /> is valid if and only if the vectors are linearly dependent. This property can be regarded as a generalization of the [[Cauchy inequality|Cauchy inequality]]:
+
1) A Gram determinant is non-negative, i.e. $  \Gamma \geq  0 $.  
 +
The equality $  \Gamma = 0 $
 +
is valid if and only if the vectors are linearly dependent. This property can be regarded as a generalization of the [[Cauchy inequality|Cauchy inequality]]:
  
<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/g/g044/g044740/g0447409.png" /></td> </tr></table>
+
$$
 +
\Gamma ( a _ {1} , a _ {2} )  \geq  0,
 +
$$
  
 
or
 
or
  
<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/g/g044/g044740/g04474010.png" /></td> </tr></table>
+
$$
 +
( a _ {1} , a _ {1} )  ( a _ {2} , a _ {2} )  \geq  \
 +
( a _ {1} , a _ {2} )  ( a _ {2} , a _ {1} )  = \
 +
| ( a _ {1} , a _ {2} ) |  ^ {2} .
 +
$$
  
 
In particular, a Gram determinant is equal to zero if any of its principal minors (which is also a Gram determinant) is zero.
 
In particular, a Gram determinant is equal to zero if any of its principal minors (which is also a Gram determinant) is zero.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474011.png" />, equality holding if and only if the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474013.png" /> are orthogonal or if one of the determinants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474015.png" /> is equal to zero. The geometrical meaning of this inequality is that the volume of the parallelotope is not larger than the product of the volumes of complementary faces. In particular,
+
2) $  \Gamma ( a _ {1} \dots a _ {n} ) \leq  \Gamma ( a _ {1} \dots a _ {p} )  \Gamma ( a _ {p + 1 }  \dots a _ {n} ) $,  
 +
equality holding if and only if the subspaces $  L ( a _ {1} \dots a _ {p} ) $
 +
and $  L ( a _ {p+} 1 \dots a _ {n} ) $
 +
are orthogonal or if one of the determinants $  \Gamma ( a _ {1} \dots a _ {p} ) $,  
 +
$  \Gamma ( a _ {p+} 1 \dots a _ {n} ) $
 +
is equal to zero. The geometrical meaning of this inequality is that the volume of the parallelotope is not larger than the product of the volumes of complementary faces. In particular,
  
<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/g/g044/g044740/g04474016.png" /></td> </tr></table>
+
$$
 +
\Gamma ( a _ {1} \dots a _ {n} )  \leq  \
 +
\Gamma ( a _ {1} ) \dots \Gamma ( a _ {n} ).
 +
$$
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474017.png" />, where
+
3) $  \Gamma ( a _ {1} \dots a _ {n} ) = \Gamma ( a _ {1} \dots a _ {n - 1 }  ) h  ^ {2} $,  
 +
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/g/g044/g044740/g04474018.png" /></td> </tr></table>
+
$$
 +
= \min _ {x  ^ {1} \dots x ^ {n - 1 } } \
 +
\left \| a _ {n} -
 +
\sum _ {i = 1 } ^ { {n }  - 1 }
 +
x  ^ {i} a _ {i} \right \|
 +
$$
  
is the distance from the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474019.png" /> to the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474020.png" />, i.e. the best quadratic approximation (cf. [[Best approximation|Best approximation]]) to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474021.png" /> by polynomials of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474022.png" />.
+
is the distance from the element $  a _ {n} $
 +
to the subspace $  L ( a _ {1} \dots a _ {n-} 1 ) $,  
 +
i.e. the best quadratic approximation (cf. [[Best approximation|Best approximation]]) to the element $  a _ {n} $
 +
by polynomials of the type $  \sum _ {i=} 1  ^ {n-} 1 x  ^ {i} a _ {i} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474023.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474024.png" />-dimensional vectors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474025.png" />, then
+
If $  a _ {1} \dots a _ {n} $
 +
are $  n $-
 +
dimensional vectors, $  a _ {i} = ( a _ {i}  ^ {1} \dots a _ {i}  ^ {n} ) $,  
 +
then
  
<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/g/g044/g044740/g04474026.png" /></td> </tr></table>
+
$$
 +
\Gamma ( a _ {1} \dots a _ {n} )  = \
 +
(  \mathop{\rm det}  | a _ {i}  ^ {j} | )  ^ {2} ,\ \
 +
i, j = 1 \dots n.
 +
$$
  
 
Gram determinants were introduced by J.P. Gram [[#References|[1]]] and, independently, by K.A. Andreev [[#References|[2]]] in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions.
 
Gram determinants were introduced by J.P. Gram [[#References|[1]]] and, independently, by K.A. Andreev [[#References|[2]]] in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions.
Line 41: Line 99:
 
The Gram determinant is a special case of determinants of the type
 
The Gram determinant is a special case of determinants of the type
  
<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/g/g044/g044740/g04474027.png" /></td> </tr></table>
+
$$
 +
\Gamma \left (
 +
 
 +
\begin{array}{c}
 +
a _ {1} \dots a _ {n}  \\
 +
b _ {1} \dots b _ {n}  \\
 +
\end{array}
 +
 
 +
\right )  =   \mathop{\rm det}  | ( a _ {i} , b _ {j} ) | ,\ \
 +
i, j = 1 \dots n,
 +
$$
  
which are Hermitian and bilinear with respect to vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474029.png" />. If all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474030.png" /> are of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044740/g04474031.png" />, then the following formula is valid:
+
which are Hermitian and bilinear with respect to vectors $  a _ {i} $
 +
and $  b _ {j} $.  
 +
If all $  a _ {i} $
 +
are of class $  L _ {2} ( E) $,  
 +
then the following formula is valid:
  
<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/g/g044/g044740/g04474032.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det}  ( a _ {i} , b _ {j} ) =
 +
$$
  
<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/g/g044/g044740/g04474033.png" /></td> </tr></table>
+
$$
 +
= \
 +
{
 +
\frac{1}{n!}
 +
} \int\limits _ { E } \dots \int\limits _ { E }  \mathop{\rm det}  | a _ {i} ( x _ {j} )  |  \mathop{\rm det} |  \overline{ {b _ {i} }}\; ( x _ {j} ) |  dx _ {1} \dots dx _ {n} .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.P. Gram,  "On Raekkeudviklinger bestemmte ved Hjaelp of de mindste Kvadraters Methode" , Copenhagen  (1879)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.A. Andreev,  , ''Selected work'' , Khar'kov  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.P. Gram,  "On Raekkeudviklinger bestemmte ved Hjaelp of de mindste Kvadraters Methode" , Copenhagen  (1879)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.A. Andreev,  , ''Selected work'' , Khar'kov  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:42, 5 June 2020


A determinant of the form

$$ \Gamma = \Gamma ( a _ {1} \dots a _ {n} ) = \ \mathop{\rm det} | ( a _ {i} , a _ {k} ) | ,\ \ i, k = 1 \dots n, $$

where $ a _ {1} \dots a _ {n} $ are elements of a (pre-)Hilbert space and $ ( a _ {i} , a _ {k} ) $ are their scalar products. A Gram determinant is equal to the square of the $ n $- dimensional volume of the parallelotope constructed on $ a _ {1} \dots a _ {n} $.

A Gram determinant is the determinant of a non-negative Hermitian form

$$ \sum _ {i, k = 1 } ^ { n } ( a _ {i} , a _ {k} ) \xi _ {i} \overline{ {\xi _ {k} }}\; = \ \left \| \sum _ {i = 1 } ^ { n } a _ {i} \xi _ {i} \ \right \| ^ {2} , $$

which determines its basic properties:

1) A Gram determinant is non-negative, i.e. $ \Gamma \geq 0 $. The equality $ \Gamma = 0 $ is valid if and only if the vectors are linearly dependent. This property can be regarded as a generalization of the Cauchy inequality:

$$ \Gamma ( a _ {1} , a _ {2} ) \geq 0, $$

or

$$ ( a _ {1} , a _ {1} ) ( a _ {2} , a _ {2} ) \geq \ ( a _ {1} , a _ {2} ) ( a _ {2} , a _ {1} ) = \ | ( a _ {1} , a _ {2} ) | ^ {2} . $$

In particular, a Gram determinant is equal to zero if any of its principal minors (which is also a Gram determinant) is zero.

2) $ \Gamma ( a _ {1} \dots a _ {n} ) \leq \Gamma ( a _ {1} \dots a _ {p} ) \Gamma ( a _ {p + 1 } \dots a _ {n} ) $, equality holding if and only if the subspaces $ L ( a _ {1} \dots a _ {p} ) $ and $ L ( a _ {p+} 1 \dots a _ {n} ) $ are orthogonal or if one of the determinants $ \Gamma ( a _ {1} \dots a _ {p} ) $, $ \Gamma ( a _ {p+} 1 \dots a _ {n} ) $ is equal to zero. The geometrical meaning of this inequality is that the volume of the parallelotope is not larger than the product of the volumes of complementary faces. In particular,

$$ \Gamma ( a _ {1} \dots a _ {n} ) \leq \ \Gamma ( a _ {1} ) \dots \Gamma ( a _ {n} ). $$

3) $ \Gamma ( a _ {1} \dots a _ {n} ) = \Gamma ( a _ {1} \dots a _ {n - 1 } ) h ^ {2} $, where

$$ h = \min _ {x ^ {1} \dots x ^ {n - 1 } } \ \left \| a _ {n} - \sum _ {i = 1 } ^ { {n } - 1 } x ^ {i} a _ {i} \right \| $$

is the distance from the element $ a _ {n} $ to the subspace $ L ( a _ {1} \dots a _ {n-} 1 ) $, i.e. the best quadratic approximation (cf. Best approximation) to the element $ a _ {n} $ by polynomials of the type $ \sum _ {i=} 1 ^ {n-} 1 x ^ {i} a _ {i} $.

If $ a _ {1} \dots a _ {n} $ are $ n $- dimensional vectors, $ a _ {i} = ( a _ {i} ^ {1} \dots a _ {i} ^ {n} ) $, then

$$ \Gamma ( a _ {1} \dots a _ {n} ) = \ ( \mathop{\rm det} | a _ {i} ^ {j} | ) ^ {2} ,\ \ i, j = 1 \dots n. $$

Gram determinants were introduced by J.P. Gram [1] and, independently, by K.A. Andreev [2] in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions.

The Gram determinant is used in many problems of linear algebra and function theory: studies of linear dependence of systems of vectors or functions, orthogonalization of systems of functions, construction of projections, and also in studies on the properties of systems of functions. See also Gram matrix.

The Gram determinant is a special case of determinants of the type

$$ \Gamma \left ( \begin{array}{c} a _ {1} \dots a _ {n} \\ b _ {1} \dots b _ {n} \\ \end{array} \right ) = \mathop{\rm det} | ( a _ {i} , b _ {j} ) | ,\ \ i, j = 1 \dots n, $$

which are Hermitian and bilinear with respect to vectors $ a _ {i} $ and $ b _ {j} $. If all $ a _ {i} $ are of class $ L _ {2} ( E) $, then the following formula is valid:

$$ \mathop{\rm det} ( a _ {i} , b _ {j} ) = $$

$$ = \ { \frac{1}{n!} } \int\limits _ { E } \dots \int\limits _ { E } \mathop{\rm det} | a _ {i} ( x _ {j} ) | \mathop{\rm det} | \overline{ {b _ {i} }}\; ( x _ {j} ) | dx _ {1} \dots dx _ {n} . $$

References

[1] J.P. Gram, "On Raekkeudviklinger bestemmte ved Hjaelp of de mindste Kvadraters Methode" , Copenhagen (1879)
[2] K.A. Andreev, , Selected work , Khar'kov (1955) (In Russian)
[3] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)

Comments

Using Gram determinants and their properties one can prove Hadamard's determinant theorem (cf. Hadamard theorem).

References

[a1] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
How to Cite This Entry:
Gram determinant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gram_determinant&oldid=18442
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article