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

The Gramian matrix can be used to test for linear independence of functions. Namely, the functions are linearly independent if and only if ${\displaystyle G}$ 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

${\displaystyle x_{1},\dots ,x_{n}}$

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

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

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.

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 ${\displaystyle x_{1},\dots ,x_{n}}$ by

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

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 ${\displaystyle P^{\top }GP}$.

See also

• Controllability Gramian
• Observability Gramian
• Overlap matrix

External links

• Barth, Nils (1999). "The Gramian and K-Volume in N-Space: Some Classical Results in Linear Algebra". Journal of Young Investigators. 2.

• Systems theory
• Matrices
• Determinants