LogmGx

LogmGx denotes the matrix logarithm of the Gram matrix associated with a collection of vectors X, written as logm(G_x). In many numerical and mathematical texts, G_x := X^T X is the Gram matrix of X under the standard inner product, where X is a d-by-n matrix whose columns are the vectors x_i in R^d. The Gram matrix is symmetric and positive semidefinite; it is positive definite if the columns of X are linearly independent, in which case all eigenvalues are strictly positive.

The matrix logarithm logm(G_x) is defined for positive definite matrices by spectral calculus: if G_x = Q Λ Q^T with Λ = diag(λ_i) and λ_i > 0, then logm(G_x) = Q diag(log λ_i) Q^T. In practice, logm is computed via eigenvalue decomposition or other SPD-friendly methods, with the symmetric positive definite property providing numerical stability. When G_x is singular (some λ_i = 0), logm is undefined; common remedies include adding a small regularization ε I to obtain G_x + ε I, or working with the pseudo-logarithm on the support of G_x.

Applications of logm(G_x) appear in areas such as covariance analysis, diffusion maps, kernel methods, and information geometry, where the log-Euclidean metric is used on the space of SPD matrices. The logarithm preserves positive definiteness and maps SPD matrices to the space of symmetric matrices, enabling linear operations in log-space.

See also: matrix logarithm, Gram matrix, positive definite matrices, log-Euclidean metric, SPD geometry.