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Gram matrix

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In systems theory and linear algebra, the Gramian matrix of a set of functions { l i ( ⋅ ) , i = 1 , … , n } {\displaystyle \{l_{i}(\cdot ),\,i=1,\dots ,n\}} is a real-valued symmetric matrix G = [ G i j ] {\displaystyle G=} , where G i j = ∫ t 0 t f l i ( τ ) l j ( τ ) d τ {\displaystyle G_{ij}=\int _{t_{0}}^{t_{f}}l_{i}(\tau )l_{j}(\tau )\,d\tau } .

The Gramian matrix can be used to test for linear independence of functions. Namely, 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

x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}}

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

( x i | x j ) {\displaystyle (x_{i}|x_{j})\,} .

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.

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 basis x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} by

G i , j = B ( x i , x j ) {\displaystyle G_{i,j}=B(x_{i},x_{j})\,} .

The matrix will be symmetric if the bilinear form B is. Under change of basis represented by an invertible matrix P, the Gram matrix will change by a matrix congruence to P ⊤ G P {\displaystyle P^{\top }GP} .

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