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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>
The functions are linearly independent if and only if <math>G</math> is ]. Its ] is known as the '''Gram determinant''' or '''Gramian'''. It is named for ]. The functions are linearly independent if and only if <math>G</math> is ]. Its ] is known as the '''Gram determinant''' or '''Gramian'''. It is named for ].

Revision as of 09:45, 6 August 2006

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 ( ⋅ ) , i = 1 , … , n } {\displaystyle \{l_{i}(\cdot ),\,i=1,\dots ,n\}} is defined as:

G = [ G i j ] , G i j = ∫ t 0 t f l i ( τ ) l j ( τ ) d τ {\displaystyle 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 {\displaystyle G} is nonsingular. Its determinant is known as the Gram determinant or Gramian. It is named for Jørgen Pedersen Gram.

In fact this is a special case of a quantitative measure of linear independence of vectors, available in any Hilbert space. According to that definition, for E a real prehilbert space, if

x1,..., xn

are n vectors of E, the associated Gram matrix is the symmetric matrix

(xi|xj).

The Gram determinant is the determinant of this matrix,

G ( x 1 , … , x n ) = | ( x 1 | x 1 ) ( x 1 | x 2 ) … ( x 1 | x n ) ( x 2 | x 1 ) ( x 2 | x 2 ) … ( x 2 | x n ) ⋮ ⋮ ⋮ ( x n | x 1 ) ( x n | x 2 ) … ( x n | x n ) | {\displaystyle G(x_{1},\dots ,x_{n})={\begin{vmatrix}(x_{1}|x_{1})&(x_{1}|x_{2})&\dots &(x_{1}|x_{n})\\(x_{2}|x_{1})&(x_{2}|x_{2})&\dots &(x_{2}|x_{n})\\\vdots &\vdots &&\vdots \\(x_{n}|x_{1})&(x_{n}|x_{2})&\dots &(x_{n}|x_{n})\end{vmatrix}}}

All eigenvalues of a Gramian matrix are real and non-negative and the matrix is thus also positive semidefinite.

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