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
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(1)
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(Ayres 1962, p. 206).
A specific example is given by
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(2)
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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
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(3)
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The convention that the submatrices have 1s on the subdiagonal instead of the superdiagonal
is also used sometimes (Faddeeva 1958, p. 50).
Portions of this entry contributed by Todd
Rowland
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.
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