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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 lambda_i. In particular, it is a block matrix of the form

 [lambda_1 1 0 ... 0; 0 lambda_1 1 ... 0; 0 0 lambda_1 ... 0; | ... ... ... 1; 0 0 0 ... lambda_1  ;  ... ;   lambda_k 1 0 ... 0; 0 lambda_k 1 ... 0; 0 0 lambda_k ... 0; | ... ... ... 1; 0 0 0 ... lambda_k]
(1)

(Ayres 1962, p. 206).

A specific example is given by

 [5 1 0 0 0 0; 0 5 1 0 0 0; 0 0 5 0 0 0; 0 0 0 1-2i 1 0; 0 0 0 0 1-2i 1; 0 0 0 0 0 1-2i],
(2)

which has three Jordan blocks. (Note that the degenerate case of a 1×1 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 A can be written in Jordan canonical form by finding a Jordan basis b_(i,j) 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 lambda can be recovered by the sequence

 a_i=dimNull(A-lambdaI)^i.
(3)

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

SEE ALSO: Jordan Basis, Jordan Block, Jordan Matrix Decomposition

Portions of this entry contributed by Todd Rowland

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.




CITE THIS AS:

Rowland, Todd and Weisstein, Eric W. "Jordan Canonical Form." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JordanCanonicalForm.html

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