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Gram matrix

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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.


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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. L.P. Kuptsov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Gram_matrix&oldid=19116
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098