Kaseïnematrices

Kaseïnematrices is a term used in a small corner of linear algebra to describe a class of square matrices that encode paired linear transformations under a designated bilinear or quadratic pairing. The concept is not yet standardized, and definitions can vary between sources, but they share a focus on how a matrix represents a map relative to a structure that links two components of a vector space.

One common variant defines a kaseïnematrix as an n×n matrix M over a field F that, with respect to a fixed basis adapted to a nondegenerate bilinear form B, has a block structure M = [A B; C D] where the off-diagonal blocks satisfy a compatibility condition, such as B being related to C by a duality induced by B (for example B being the transpose of C under B’s pairing). In such setups, choosing B, A, D to respect the pairing can yield matrices that preserve a Gram form or interact nicely with the associated bilinear structure. A widely cited special case recovers familiar matrix classes: symmetric matrices, persymmetric (block-reversal symmetric) matrices, and certain Toeplitz-plus-Hankel types can appear as kaseïnematrices under appropriate choices of pairing and basis.

Properties frequently studied include spectral behavior under the pairing, invariance under transformations that preserve the bilinear form, and how the block structure affects numerical methods. Relationships with symmetric and block matrices, as well as with bilinear forms and dualities, are central themes.

Because the term appears in limited or evolving literature, the exact definition and scope of kaseïnematrices may differ across texts. See also symmetric matrices, block matrices, bilinear forms, and Toeplitz/Hankel matrices for related concepts.