PlanetMath: Jordan canonical form theorem

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Jordan canonical form theorem

(Theorem)

A Jordan block or Jordan matrix is a matrix of the form

$\displaystyle \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\ 0 & \lambda & 1 &... ...ts & \vdots & \vdots & \ddots & 1\ 0 & 0 & 0 & \cdots & \lambda \end{pmatrix}$

with a constant value $\lambda$ along the diagonal and 1's on the superdiagonal. Some texts place the 1's on the subdiagonal instead.

Theorem 1 Let be a finite-dimensional vector space over a field and $t:V \to V$ be a linear transformation. Then, if the characteristic polynomial factors completely over , there will exist a basis of with respect to which the matrix of is of the form

$\displaystyle \begin{pmatrix} J_{1} & 0 & \cdots& 0\ 0 & J_{2} & \cdots & 0\ & & \cdots & \ 0 & 0 & \cdots & J_{k} \end{pmatrix}$

where each $J_{i}$ is a Jordan block in which $\lambda = \lambda_{i}$ .

The matrix in Theorem 1 is called a Jordan canonical form for the transformation t.

"Jordan canonical form theorem" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Other names:

Jordan canonical form

Also defines:

Jordan block, Jordan matrix

Attachments:: proof of Jordan canonical form theorem (Proof) by CWoo

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Cross-references: transformation, basis, factors, characteristic polynomial, linear transformation, field, vector space, finite-dimensional, diagonal, matrix
There are 11 references to this entry.

This is version 13 of Jordan canonical form theorem, born on 2002-08-26, modified 2007-11-06.
Object id is 3364, canonical name is JordanCanonicalFormTheorem.
Accessed 18385 times total.

Classification:

AMS MSC:

15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)

Pending Errata and Addenda

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