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'Jordan canonical form theorem'
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| Title of object: |
Jordan canonical form theorem |
| Canonical Name: |
JordanCanonicalFormTheorem |
| Type: |
Theorem |
| Created on: |
2002-08-26 10:37:19 |
| Modified on: |
2007-11-05 09:39:35 |
| Classification: |
msc:15A18 |
| Defines: |
Jordan block, Jordan matrix |
| Synonyms: |
Jordan canonical form theorem=Jordan canonical form |
Revision comment (for changes between this and next version):
| Changes for correction #13230 ('\cdots'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newtheorem*{thms}{Theorem} |
Content:
A \textbf{Jordan block} or \textbf{Jordan matrix} is a matrix of the form
$$\begin{pmatrix}
\lambda & 1 & 0 & ... & 0\\
0 & \lambda & 1 & ... & 0\\
0 & 0 & \lambda & ... & 0\\
\vdots & \vdots & \vdots & \ddots & 1\\
0 & 0 & 0 & ... & \lambda
\end{pmatrix}$$
with a constant value $\lambda$ along the diagonal and 1's on the superdiagonal. Some texts \PMlinkescapetext{place} the 1's on the subdiagonal instead.
\begin{thms} 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
$$\begin{pmatrix}
J_{1} & 0 & ... & 0\\
0 & J_{2} & ... & 0\\
& & ... & \\
0 & 0 & ... & J_{k}
\end{pmatrix}$$
where each $J_{i}$ is a Jordan block in which $\lambda = \lambda_{i}$.
\end{thms}
The matrix in Theorem 1 is called a \emph{Jordan canonical form} for the transformation \emph{t}.
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