are called a right eigenvector and a left eigenvector,
respectively, where
is the transpose of
and
is its conjugate-transpose.
The word eigenvector alone will mean right eigenvector.
Since there are many kinds of eigenproblems, and associated algorithms, we propose some simple top level categories to help classify them. The ultimate decision tree presented to the reader will begin with easier concepts and questions about the eigenproblem in an attempt to classify it, and proceed to harder questions. For the purposes of this overview, we will use rather more advanced categories in order to be brief but precise. For background, see [12][25][23].
Regular pencils have well-defined sets of eigenvalues
which change continuously as functions of and
;
this is a minimal requirement to be able to compute the
eigenvalues accurately, in the absence of other
constraints on
and
. Singular pencils have eigenvalues
which can change discontinuously as functions of
and
; extra information about
and
,
as well as special algorithms which use this information,
are necessary in order to compute meaningful eigenvalues.
Regular and singular pencils have correspondingly different
canonical forms representing their spectral
decompositions. The Jordan Canonical Form of a single
matrix is the best known; the Kronecker Canonical Form
of a singular pencil is the most general. More will be
discussed in section 4.3
below.
It is sometimes possible to change the classification of an eigenproblem
by simple transformations. For example, multiplying a skew-Hermitian
matrix (i.e.,
)
by the constant
makes it Hermitian, so its
eigenproblem becomes self-adjoint. Such simple transformations are very
useful in finding the best algorithm for a problem.
Many other definition and notation will be introduced in section 4.3.