Matrixpaar

Matrixpaar is a term used in German-language mathematics to denote a pair of matrices (A, B) of the same size. Such a pair defines a matrix pencil A − λB, where λ is a scalar parameter. The study of matrix pairs encompasses a range of problems in linear algebra, most notably the generalized eigenvalue problem Av = λBv, which seeks nonzero vectors v and corresponding scalars λ that satisfy the equation.

If the pencil is regular, i.e., det(A − λB) is not identically zero, there exists a finite set of generalized eigenvalues and corresponding eigenvectors. The case B = I reduces the problem to ordinary eigenvalues of A. Matrix pairs also arise in applications such as structural dynamics, vibration analysis, control theory, signal processing, and the modeling of differential-algebraic equations, where A and B encode system relations and constraints.

Canonical forms and classification play a central role. The Kronecker canonical form provides a complete classification of matrix pencils up to strict equivalence, separating finite and infinite eigenvalues and identifying singular blocks. For computation, numerical linear algebra offers algorithms to solve generalized eigenvalue problems, notably the QZ algorithm (generalized Schur form) for regular pencils, and specialized methods that preserve symmetry or definiteness in symmetric definite pencils.

See also: generalized eigenvalue problem, matrix pencil, Kronecker canonical form, QZ algorithm.