Gram matrix - Wikipedia

Abstraction: Matrix of pairwise inner products encoding geometric structure of a vector set

Key points:

• G_ij = <v_i, v_j>; for real vectors G = V^T V where columns of V are the vectors; for complex vectors G = V^* V
• Always positive semidefinite (PSD); positive definite if and only if the vectors are linearly independent
• Every PSD matrix is a Gram matrix (converse); diagonalization of Gram matrix is the SVD; rank equals dimension of the vectors' span
• Gram determinant (det G) equals the square of the n-dimensional volume of the parallelotope spanned by the vectors; nonzero iff vectors are linearly independent
• Vectors realizing a Gram matrix are unique up to orthogonal (or unitary) transformations
• Applications: kernel functions in ML (kernel PCA), finite element basis overlap matrices, controllability/observability Gramians in control theory, covariance matrices for centered random variables

Connections: Linear Algebra · Kernel Methods · Positive Semidefinite Matrices

Source: http://en.wikipedia.org/wiki/Gramian_matrix