Gram matrix: Difference between revisions

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	In ] and ], a '''Gramian matrix''' is a ] ] that can be used to test for ] of ]s. The Gramian matrix of a set of functions <math>\{l_i(\cdot),\,i=1,\dots,n\}</math> is defined as		In ] and ], a '''Gramian matrix''' is a ] ] that can be used to test for ] of ]s. The Gramian matrix of a set of functions <math>\{l_i(\cdot),\,i=1,\dots,n\}</math> is defined as
	:<math>G=,\,\,G_{ij}=\int_{t_0}^{t_f} l_i(\tau)l_j(\tau)\, d\tau </math>		:<math>G=,\,\,G_{ij}=\int_{t_0}^{t_f} l_i(\tau)l_j(\tau)\, d\tau </math>
	~~If the~~ functions are linearly independent, ~~then~~ <math>G</math> is ]. Its ] is known as the '''Gram determinant''' or '''Gramian'''.		The functions are linearly independent if and only if <math>G</math> is ]. Its ] is known as the '''Gram determinant''' or '''Gramian'''.

	In fact this is a special case of a quantitative measure of linear independence of vectors, available in any ].		In fact this is a special case of a quantitative measure of linear independence of vectors, available in any ].

Revision as of 15:16, 4 November 2005

In systems theory and linear algebra, a Gramian matrix is a real-valued symmetric matrix that can be used to test for linear independence of functions. The Gramian matrix of a set of functions $\{l_{i}(\cdot ),\,i=1,\dots ,n\}$ is defined as

G=,\,\,G_{ij}=\int _{t_{0}}^{t_{f}}l_{i}(\tau )l_{j}(\tau )\,d\tau

The functions are linearly independent if and only if $G$ is nonsingular. Its determinant is known as the Gram determinant or Gramian.

In fact this is a special case of a quantitative measure of linear independence of vectors, available in any Hilbert space.