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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.

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

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

• Systems theory
• Matrices