gramian

Gramian refers to a Gram matrix in linear algebra: a matrix of inner products for a set of vectors. Given vectors v1, …, vk in an inner product space, the Gram matrix G has entries G_ij = ⟨v_i, v_j⟩. If A is the n-by-k matrix whose columns are the vectors, then G = A^T A (A* A in complex spaces). The Gram matrix is symmetric (Hermitian in the complex case) and positive semidefinite. Its rank equals the dimension of the span of the vectors, and its determinant, the Gram determinant, equals the square of the volume of the parallelepiped spanned by the vectors. In particular, det(G) = 0 if and only if the vectors are linearly dependent.

Gram matrices appear in many contexts. They encode inner products and pairwise similarities, and their eigenvalues reflect directions of maximal variance or independence. In statistics and data analysis, the Gram matrix of centered data X (with observations as rows and variables as columns) is X^T X, and it relates to the covariance matrix up to a scaling. In numerical linear algebra, computing G via G = A^T A is common, and Cholesky or other factorizations are typically used when G is positive definite.

Beyond the standard Gram matrix, the term Gramian is also used in control theory to denote reachability and observability measures. For a continuous-time system x' = Ax + Bu, the controllability Gramian Wc(T) = ∫_0^T e^{A t} B B^T e^{A^T t} dt captures the energy needed to move the state; if Wc(T) is positive definite, the system is controllable on [0, T]. An observability Gramian plays a related role for sensing. See also Gram matrix, inner product, and covariance.