Misplaced Pages

Revision as of 23:25, 14 May 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 ${\displaystyle \{l_{i}(\cdot ),\,i=1,\dots ,n\}}$ is defined as

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

The functions are linearly independent if and only if ${\displaystyle G}$ 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,..., x_n

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

(x_i|x_j).

The Gram determinant is the determinant of this matrix,

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

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

• Systems theory
• Matrices