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Jordan canonical form theorem
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(Theorem)
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A Jordan block or Jordan matrix is a matrix of the form
$$\begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \cdots & 0\\ 0 & 0 & \lambda & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & 1\\ 0 & 0 & 0 & \cdots & \lambda \end{pmatrix}$$
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 $V$ be a finite-dimensional vector space over a field $F$ and $t:V \to V$ be a linear transformation. Then, if the characteristic polynomial factors completely over $F$ , there will exist a basis of $V$ with respect to which the matrix of $t$ is of the form
$$\begin{pmatrix} J_{1} & 0 & \cdots& 0\\ 0 & J_{2} & \cdots & 0\\ & & \cdots & \\ 0 & 0 & \cdots & J_{k} \end{pmatrix}$$
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.
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Cross-references: transformation, theorem, 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 23229 times total.
Classification:
| AMS MSC: | 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors) |
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Pending Errata and Addenda
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