Gramians

Gramians are families of matrices that arise from inner product structures in linear algebra and control theory. They encode pairwise inner products among vectors, or energy transfer properties of dynamical systems, and play a central role in measuring independence, reachability, and similarity.

For a set of vectors v1,...,vn in a real or complex inner product space, the Gram matrix G is defined by G_ij = vi · vj. This matrix is symmetric and positive semidefinite; its rank equals the dimension of the span of the vectors. The determinant det(G) equals the square of the volume of the parallelepiped spanned by the vectors, and G is singular precisely when the vectors are linearly dependent.

In machine learning and statistics, a kernel Gram matrix K is defined by K_ij = k(x_i, x_j) for a positive semidefinite kernel k. Such Gram matrices are symmetric and PSD and underpin kernel-based methods, including support vector machines, kernel PCA, and Gaussian processes; their eigenvalues reveal data geometry and capacity.

In control theory, controllability and observability Gramian matrices quantify how much energy input or state information reaches the system. For a continuous-time linear system x' = Ax + Bu, the controllability Gramian Wc = ∫0^∞ e^{At} B B^T e^{A^T t} dt; for discrete-time, Wc = Σ A^k B B^T (A^T)^k. The observability Gramian Wo is defined similarly with C^T C. These Gramians satisfy Lyapunov equations: A Wc + Wc A^T + B B^T = 0; A^T Wo + Wo A + C^T C = 0; they underpin model reduction via balanced truncation.

Gramians are typically positive semidefinite, enabling interpretations via eigenvalues and singular values. They may be ill-conditioned for nearly dependent vectors or large systems, prompting regularization or low-rank approximations. The term also encompasses determinant-based volume measures and kernel matrices in data analysis.