Difference between revisions of "Gram matrix"
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+ | $#C+1 = 13 : ~/encyclopedia/old_files/data/G044/G.0404750 Gram matrix | ||
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The square matrix | The square matrix | ||
− | + | $$ | |
+ | G ( a _ {1} \dots a _ {k} ) = \ | ||
+ | \| g _ {\alpha \beta } \| , | ||
+ | $$ | ||
− | consisting of pairwise scalar products | + | 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 | + | 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 | 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|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) |
Gram matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gram_matrix&oldid=35177