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In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by . For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply (or for complex vectors using the conjugate transpose), where ''V'' is a matrix whose columns are the vectors . An important application is to compute linear independence: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. ==Examples== Most commonly, the vectors are elements of a Euclidean space, or are functions in an ''L''2 space, such as continuous functions on a compact interval () (which are a subspace of ''L'' 2(())). Given real-valued functions on the interval , the Gram matrix , is given by the standard inner product on functions: : Given a real matrix ''A'', the matrix ''A''T''A'' is a Gram matrix (of the columns of ''A''), while the matrix ''AA''T is the Gram matrix of the rows of ''A''. For a general bilinear form ''B'' on a finite-dimensional vector space over any field we can define a Gram matrix ''G'' attached to a set of vectors by . The matrix will be symmetric if the bilinear form ''B'' is symmetric. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gramian matrix」の詳細全文を読む スポンサード リンク
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