Kaseïnematrix

Kaseïnematrix is a term used in some theoretical discussions to denote a class of square matrices whose entries are defined by a kernel function or a sequence, rather than by a fixed algebraic rule. There is no single standard definition, and the name does not correspond to a widely recognized mathematical object. In its broad sense, a kaseïnematrix A of size n can be described by A[i, j] = k(i, j) for a function k: N × N → F, where F is a field such as the real or complex numbers. If k is symmetric and positive semidefinite, A qualifies as a kernel (Gram-type) matrix, interpretable as inner products of feature vectors in some Hilbert space.

Common constructions use different kernel choices. For example, k(i, j) = i · j yields a rank-one matrix with a single nonzero eigenvalue equal to the square of the sum of indices; k(i, j) = min(i, j) produces a matrix connected to cumulative sums and has specific structural properties; k(i, j) = exp(-|i − j|) gives a symmetric Toeplitz matrix that can be positive definite under suitable conditions. In many treatments, the kernel is chosen to reflect data structure or operator discretization, resulting in symmetry, potential low-rank structure, or particular sparsity patterns.

Applications and properties commonly associated with kaseïnematrices include spectral analysis, kernel methods in machine learning, time-series modeling, and discretizations of integral operators. Computational considerations focus on conditioning, efficiency of eigen decompositions, and scalable approximations for large n.

The etymology of the term is not standardized; it appears in some writings as a placeholder or as a name chosen by authors for a family of kernel-based matrices. See also: Gram matrix, kernel matrix, Toeplitz matrix, Hankel matrix.