In linear algebra, if
Contents
A particular case is a linear matrix pencil
A pencil is called regular if there is at least one value of
Applications
Matrix pencils play an important role in numerical linear algebra. The problem of finding the eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, which is an implicit version of the QR algorithm to solve the associated eigenvalue problem
Pencil generated by commuting matrices
If
- consists only of matrices similar to a diagonal matrix, or
- has no matrices in it similar to a diagonal matrix, or
- has exactly one matrix in it similar to a diagonal matrix.