Editing Jordan normal form (section)

{{short description|Form of a matrix indicating its eigenvalues and their algebraic multiplicities}}
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<math>
\begin{pmatrix}
 {\color{red}\ulcorner}\lambda_1 1\hphantom{\lambda_1\lambda_1}{\color{red}\urcorner}\hphantom{\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1}\lambda_1 1\hphantom{\lambda_1\urcorner\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 {\color{red}\llcorner}\hphantom{\lambda_1 1\lambda_1}\lambda_1{\color{red}\lrcorner}\hphantom{\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1\lambda_1 1\lambda_1\urcorner}{\color{red}\ulcorner}\lambda_2 1\hphantom{n}{\color{red}\urcorner}\hphantom{[\lambda_3]\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1\lambda_1 1\lambda_1\lrcorner}{\color{red}\llcorner}\hphantom{\lambda_2 }\lambda_2{\color{red}\lrcorner}\hphantom{[\lambda_3]\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1\lambda_1 1\lambda_1\urcorner\ulcorner\lambda_2 1\lambda_2\urcorner}{\color{red}[}\lambda_3{\color{red}]}\hphantom{\ddots\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1\lambda_1 1\lambda_1\urcorner\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]}\ddots\hphantom{\ulcorner\lambda_n 1\lambda_n\urcorner}\\
 \hphantom{\ulcorner\lambda_1 1\lambda_1 1\lambda_1\urcorner\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]\ddots}{\color{red}\ulcorner}\lambda_n 1\hphantom{n}{\color{red}\urcorner}\\
 \hphantom{\llcorner\lambda_1 1\lambda_1 1\lambda_1\urcorner\ulcorner\lambda_2 1\lambda_2\urcorner[\lambda_3]\ddots}{\color{red}\llcorner}\hphantom{\lambda_n}\lambda_n{\color{red}\lrcorner}
\end{pmatrix}
</math>{{Clear}}Example of a matrix in Jordan normal form. <br />All matrix entries not shown are zero. The <br />outlined squares are known as "Jordan blocks". <br />Each Jordan block contains one number ''λ<sub>i</sub>'' <br />on its main diagonal, and 1s directly above <br />the main diagonal. The ''λ<sub>i</sub>''s are the eigenvalues <br />of the matrix; they need not be distinct.</div>
In [[linear algebra]], a '''Jordan normal form''', also known as a '''Jordan canonical form''',<ref>
Shilov defines the term ''Jordan canonical form'' and in a footnote says that ''Jordan normal form'' is synonymous.
These terms are sometimes shortened to ''Jordan form''. (Shilov)
The term ''Classical canonical form'' is also sometimes used in the sense of this article. (James & James, 1976)
</ref><ref name="Holt 2009 9">{{harvtxt|Holt|Rumynin|2009|p=9}}</ref>
is an [[upper triangular matrix]] of a particular form called a [[Jordan matrix]] representing a [[linear operator]] on a [[finite-dimensional]] [[vector space]] with respect to some [[Basis (linear algebra)|basis]]. Such a matrix has each non-zero off-diagonal entry equal to&nbsp;1, immediately above the main diagonal (on the [[superdiagonal]]), and with identical diagonal entries to the left and below them.

Let ''V'' be a vector space over a [[field (mathematics)|field]] ''K''. Then a basis with respect to which the matrix has the required form exists [[if and only if]] all [[eigenvalue]]s of the matrix lie in ''K'', or equivalently if the [[characteristic polynomial]] of the operator splits into linear factors over ''K''. This condition is always satisfied if ''K'' is [[algebraically closed]] (for instance, if it is the field of [[complex number]]s). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the [[algebraic multiplicity]] of the eigenvalue.<ref name="Beauregard 1973 310–316">{{harvtxt|Beauregard|Fraleigh|1973|pp=310–316}}</ref><ref name="Golub 1996 354">{{harvtxt|Golub|Van Loan|1996|p=355}}</ref><ref name="Nering 1970 118–127">{{harvtxt|Nering|1970|pp=118–127}}</ref>

If the operator is originally given by a [[square matrix]] ''M'', then its Jordan normal form is also called the Jordan normal form of ''M''. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given ''M'' is not entirely unique, as it is a [[block diagonal matrix]] formed of [[Jordan block]]s, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.<ref name="Beauregard 1973 310–316"/><ref name="Golub 1996 354"/><ref name="Nering 1970 118–127"/>

The [[Jordan–Chevalley decomposition]] is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for [[diagonalizable]] matrices, for instance [[normal matrix|normal matrices]], is a special case of the Jordan normal form.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=270–274}}</ref><ref>{{harvtxt|Golub|Van Loan|1996|p=353}}</ref><ref>{{harvtxt|Nering|1970|pp=113–118}}</ref>

The Jordan normal form is named after [[Camille Jordan]], who first stated the Jordan decomposition theorem in 1870.<ref name="Brechenmacher-thesis">Brechenmacher, [https://tel.archives-ouvertes.fr/tel-00142786 "Histoire du théorème de Jordan de la décomposition matricielle (1870-1930). Formes de représentation et méthodes de décomposition"], Thesis, 2007</ref>