Difference between revisions of "Gram matrix"
From Encyclopedia of Mathematics
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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/g/g044/g044750/g0447509.png" /></td> </tr></table> | <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> | ||
| − | 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]]. | + | 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]]. |
Revision as of 08:10, 30 November 2014
The square matrix
 |
consisting of pairwise scalar products 
of elements (vectors) of a (pre-)Hilbert space. All Gram matrices are non-negative definite. The matrix is positive definite if 
are linearly independent. The converse is also true: Any non-negative (positive) definite 
-matrix is a Gram matrix (with linearly independent defining vectors).
If 
are 
-dimensional vectors (columns) of an 
-dimensional Euclidean (Hermitian) space with the ordinary scalar product
 |
then
 |
where 
is the 
-matrix consisting of the columns 
. The symbol 
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
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