Eigendecompositions.

We discuss two eigendecompositions, or pencils that are equivalent to $A - \lambda B$, and for which it is simpler to solve eigenproblems. The decompositions are discussed in more detail in §8.7.

There is no analogy to the diagonal form; singular pencils are too complicated for this. There are only the Kronecker form and the generalized Schur-staircase form, which is sometimes also called the GUPTRI form (for generalized upper triangular form) or Kronecker-Schur form. The Kronecker form generalizes the Jordan and the Weierstrass forms to singular pencils. In addition to their discontinuities and other numerical difficulties, it adds its own difficulties associated with singular pencils. The generalized Schur-staircase form generalized both the generalized Schur form and the Weierstrass-Schur form. It can be computed stably with orthogonal transformations, but retains the discontinuities inherent in singular pencils.

Given the difficulty associated with singular pencils, partial eigendecompositions are usually not computed; all known algorithms compute essentially complete decompositions.