Grammatriisi

Grammatriisi, commonly known as the Gram matrix in English, is a matrix that encodes the inner products between a set of vectors in a real or complex inner product space. Given vectors x1, x2, ..., xn, the grammatriisi G has entries G_ij = ⟨x_i, x_j⟩, where ⟨.,⟩ denotes the inner product. If the vectors are arranged as columns of a matrix X, then G = X^T X in the real-valued case (G = X*X with the conjugate transpose in the complex case).

Properties of grammatriisi include symmetry (or Hermitian symmetry in the complex case) and positive semidefiniteness. The rank of G equals the dimension of the span of the vectors, and G is invertible exactly when the vectors are linearly independent. The diagonal entries are the squared norms of the vectors, while the off-diagonal entries measure pairwise similarity via the inner product. The determinant of G, known as the Gram determinant, equals the squared volume of the parallelepiped spanned by the vectors and provides a criterion for linear independence.

Applications of grammatriisi span several areas. In data analysis and machine learning, Gram matrices capture the inner-product structure of data and underpin kernel methods, kernel PCA, and various similarity-based algorithms. In statistics and multivariate analysis, they arise in covariance-free formulations and in computations that rely on inner products rather than explicit coordinates. The spectrum and conditioning of the Gram matrix inform about redundancy and numerical stability.

History and naming trace back to Jørgen Pedersen Gram, who studied such matrices in the 19th century. See also: Gram determinant, inner product, kernel methods, and Gram matrix in related contexts.