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Gramian : ウィキペディア英語版
Gramian matrix
In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by G_=\langle v_i, v_j \rangle. For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply G = V^\mathrm V (or G = V^\dagger V for complex vectors using the conjugate transpose), where ''V'' is a matrix whose columns are the vectors v_k.
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 G=(), is given by the standard inner product on functions:
: G_=\int_^ \ell_i(\tau)\bar(\tau)\, d\tau.
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 v_1,\dots, v_n by G_ = B(v_i,v_j) \, . 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)
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