Examples
Nilpotent matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group. From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes where is a partition of n. Thus in the case n=2 there are two nilpotent orbits, the zero orbit consisting of the zero matrix and corresponding to the partition (1,1) and the principal orbit consisting of all non-zero matrices A with zero trace and determinant,
- with
corresponding to the partition (2). Geometrically, this orbit is a two-dimensional complex quadratic cone in four dimensional vector space of matrices minus its apex.
The complex special linear group is a subgroup of the general linear group with the same nilpotent orbits. However, if we replace the complex special linear group with the real special linear group, new nilpotent orbits may arise. In particular, for n=2 there are now 3 nilpotent orbits: the zero orbit and two real half-cones (without the apex), corresponding to positive and negative values of in the parametrization above.
Read more about this topic: Nilpotent Orbit
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)