Jordan Canonical Form

The Jordan canonical form, also called the classical canonical form, of a special type of block matrix in which each block consists of Jordan blocks with possibly differing constants . In particular, it is a block matrix of the form

(1)

(Ayres 1962, p. 206).

A specific example is given by

(2)

which has three Jordan blocks. (Note that the degenerate case of a  matrix is considered a Jordan block even though it lacks a superdiagonal to be filled with 1s; cf. Strang 1988, p. 454).

Any complex matrix  can be written in Jordan canonical form by finding a Jordan basis  for each Jordan block. In fact, any matrix with coefficients in an algebraically closed field can be put into Jordan canonical form. The dimensions of the blocks corresponding to the eigenvalue  can be recovered by the sequence

(3)

The convention that the submatrices have 1s on the subdiagonal instead of the superdiagonal is also used sometimes (Faddeeva 1958, p. 50).

REFERENCES:

Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, 1962.

Faddeeva, V. N. Computational Methods of Linear Algebra. New York: Dover, p. 50, 1958.

Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.