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The square matrix
 
The square matrix
  
<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/g044750/g0447501.png" /></td> </tr></table>
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$$
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G ( a _ {1} \dots a _ {k} )  = \
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\| g _ {\alpha \beta }  \| ,
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$$
  
consisting of pairwise scalar products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447502.png" /> of elements (vectors) of a (pre-)Hilbert space. All Gram matrices are non-negative definite. The matrix is positive definite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447503.png" /> are linearly independent. The converse is also true: Any non-negative (positive) definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447504.png" />-matrix is a Gram matrix (with linearly independent defining vectors).
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consisting of pairwise scalar products $  g _ {\alpha \beta }  = ( a _  \alpha  , a _  \beta  ) $
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of elements (vectors) of a (pre-)Hilbert space. All Gram matrices are non-negative definite. The matrix is positive definite if $  a _ {1} \dots a _ {k} $
 +
are linearly independent. The converse is also true: Any non-negative (positive) definite $  ( k \times k) $-
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matrix is a Gram matrix (with linearly independent defining vectors).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447505.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447506.png" />-dimensional vectors (columns) of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g0447507.png" />-dimensional Euclidean (Hermitian) space with the ordinary scalar product
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If $  a _ {1} \dots a _ {k} $
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are $  n $-
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dimensional vectors (columns) of an $  n $-
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dimensional Euclidean (Hermitian) space with the ordinary scalar product
  
<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/g044750/g0447508.png" /></td> </tr></table>
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$$
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( a, b)  = \
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\sum _ {i = 1 } ^ { n }
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a  ^ {i} b  ^ {i} \  \left ( = \
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\sum _ {i = 1 } ^ { n }
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a  ^ {i} \overline{ {b  ^ {i} }}\; \right ) ,
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$$
  
 
then
 
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/g044750/g0447509.png" /></td> </tr></table>
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$$
 
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G ( a _ {1} \dots a _ {k} )  = \
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g04475010.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g04475011.png" />-matrix consisting of the columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g04475012.png" />. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044750/g04475013.png" /> denotes the operation of matrix transposition, while the bar denotes complex conjugation of the variable. See also [[Gram determinant|Gram determinant]].
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\overline{A}\; {}  ^ {T} A,
 
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$$
  
 +
where  $  A $
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is the  $  ( n \times k) $-
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matrix consisting of the columns  $  a _ {1} \dots a _ {k} $.
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The symbol  $  {}  ^ {T} $
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denotes the operation of [[matrix transposition]], while the bar denotes complex conjugation of the variable. See also [[Gram determinant|Gram determinant]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Introduction to linear algebra and the theory of matrices" , Noordhoff  (1950)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Introduction to linear algebra and the theory of matrices" , Noordhoff  (1950)  (Translated from German)</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


The square matrix

$$ G ( a _ {1} \dots a _ {k} ) = \ \| g _ {\alpha \beta } \| , $$

consisting of pairwise scalar products $ g _ {\alpha \beta } = ( a _ \alpha , a _ \beta ) $ of elements (vectors) of a (pre-)Hilbert space. All Gram matrices are non-negative definite. The matrix is positive definite if $ a _ {1} \dots a _ {k} $ are linearly independent. The converse is also true: Any non-negative (positive) definite $ ( k \times k) $- matrix is a Gram matrix (with linearly independent defining vectors).

If $ a _ {1} \dots a _ {k} $ are $ n $- dimensional vectors (columns) of an $ n $- dimensional Euclidean (Hermitian) space with the ordinary scalar product

$$ ( a, b) = \ \sum _ {i = 1 } ^ { n } a ^ {i} b ^ {i} \ \left ( = \ \sum _ {i = 1 } ^ { n } a ^ {i} \overline{ {b ^ {i} }}\; \right ) , $$

then

$$ G ( a _ {1} \dots a _ {k} ) = \ \overline{A}\; {} ^ {T} A, $$

where $ A $ is the $ ( n \times k) $- matrix consisting of the columns $ a _ {1} \dots a _ {k} $. The symbol $ {} ^ {T} $ denotes the operation of matrix transposition, while the bar denotes complex conjugation of the variable. See also Gram determinant.

Comments

References

[a1] H. Schwerdtfeger, "Introduction to linear algebra and the theory of matrices" , Noordhoff (1950) (Translated from German)
How to Cite This Entry:
Gram matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gram_matrix&oldid=19116
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article