Matrix pencil

In linear algebra, if formula_1 are formula_2 complex matrices for some nonnegative integer formula_3, and formula_4 (the zero matrix), then the matrix pencil of degree formula_3 is the matrix-valued function defined on the complex numbers formula_6
A particular case is a linear matrix pencil formula_7 with formula_8 where formula_9 and formula_10 are complex (or real) formula_11 matrices. We denote it briefly with the notation formula_12.
A pencil is called "regular" if there is at least one value of formula_13 such that formula_14. We call "eigenvalues" of a matrix pencil formula_12 all complex numbers formula_13 for which formula_17 (see eigenvalue for comparison). The set of the eigenvalues is called the "spectrum" of the pencil and is written formula_18.
Moreover, the pencil is said to have one or more eigenvalues at infinity if formula_10 has one or more 0 eigenvalues.
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 formula_20 without forming explicitly the matrix formula_21 (which could be impossible or ill-conditioned if formula_10 is singular or near-singular)
Pencil generated by commuting matrices.
If formula_23, then the pencil generated by formula_9 and formula_10: