Grundmatrix

Grundmatrix is a term encountered in some German-language mathematical writings to denote a foundational or base matrix from which other structures are derived. The exact meaning of the term is not standardized in English-language reference works, and its interpretation can vary by author and context.

One common usage is as a basis matrix: a Grundmatrix is a matrix whose columns form a basis for a given finite-dimensional vector space. In this sense, the Grundmatrix provides a compact representation of the basis, and it is invertible when the basis vectors are linearly independent. Changing the basis corresponds to multiplying by a change-of-basis matrix, which some authors may also refer to as a Grundmatrix in their discussions.

A second usage appears in the theory of linear differential equations and is related to the concept of a fundamental matrix. In this context, a Grundmatrix can refer to a fundamental matrix Phi(t), a matrix-valued function satisfying d/dt Phi(t) = A Phi(t) with Phi(0) = I. The fundamental matrix encodes the evolution of the homogeneous system x' = A x, with solutions of the form x(t) = Phi(t) c for constant vector c. On intervals where A is continuous, Phi(t) is invertible for all t in the interval.

Examples and notes: the 3x3 identity matrix I is a Grundmatrix for the space R^3 when viewed as a basis matrix. For a constant A, Phi(t) = e^{At} is the corresponding fundamental matrix in the differential-equation sense. Because Grundmatrix is not a universally standardized term, readers should check the surrounding text to determine which meaning is intended. See also Basis, Fundamental matrix, Change of basis.