Gram matrix

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)