# Gram matrix

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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.

#### 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=47114
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article