User:Maximilian Janisch/latexlist/Algebraic Groups/Jordan matrix

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also Jordan canonical form, Jordan normal form

A square block-diagonal matrix $j$ over a field $k$ of the form

\begin{equation} J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right| \end{equation}

where $J _ { m } ( \lambda )$ is a square matrix of order $m$ of the form

\begin{equation} J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ] \end{equation}

$\lambda \in k$. The matrix $J _ { m } ( \lambda )$ is called the Jordan block of order $m$ with eigen value $2$. Every block is defined by an elementary divisor (cf. Elementary divisors, see [5]).

For an arbitrary square matrix $4$ over an algebraically closed field $k$ there always exists a square non-singular matrix $C$ over $k$ such that $C ^ { - 1 } A C$ is a Jordan matrix (in other words, $4$ is similar over $k$ to a Jordan matrix). This assertion is valid under weaker restrictions on $k$: For a matrix $4$ to be similar to a Jordan matrix it is necessary and sufficient that $k$ contains all roots of the minimum polynomial of $4$. The matrix $C ^ { - 1 } A C$ mentioned above is called a Jordan form (or Jordan normal form) of the matrix $4$. C. Jordan [1] was one of the first to consider such a normal form (see also the historical survey in Chapts. VI and VII of [2]).

The Jordan form of a matrix is not uniquely determined, but only up to the order of the Jordan blocks. More exactly, two Jordan matrices are similar over $k$ if and only if they consist of the same Jordan blocks and differ only in the distribution of the blocks along the main diagonal. The number $C _ { m } ( \lambda )$ of Jordan blocks of order $m$ with eigen value $2$ in a Jordan form of a matrix $4$ is given by the formula

\begin{equation} C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } + \end{equation}

\begin{equation} + \operatorname { rk } ( A - \lambda E ) ^ { m + 1 } \end{equation}

where $k$ is the unit matrix of the same order $12$ as $4$, $r k B$ is the rank of the matrix $B$, and $\operatorname { rk } ( A - \lambda E ) ^ { 0 }$ is $12$, by definition.

There are other types of normal forms of matrices besides a Jordan normal form. They are resorted to, for example, when it is desired to avoid the non-uniqueness of the reduction to a Jordan normal form, or when the ground field does not contain all roots of the minimum polynomial of the matrix (see [2]–[5]).

From the point of view of the theory of invariants, a Jordan matrix is a canonical representative in the orbits of the adjoint representation of the general linear group. The determination of analogous representatives for an arbitrary reductive algebraic group is still (1978) not completely solved (see [6]–[7]).

References

[1]	C. Jordan, "Traité des substitutions et des équations algébriques" , Paris (1870) pp. 114–125 MR1188877 MR0091260 Zbl 03.0042.02
[2]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR0643362 Zbl 1139.12001
[3]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) MR1657129 MR0107649 MR0107648 Zbl 0927.15002 Zbl 0927.15001 Zbl 0085.01001
[4]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[5]	A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) Zbl 0396.15001
[6]	A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201
[7]	R. Steinberg, "Classes of elements of semisimple algebraic groups" , Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 277–283 MR0238856 Zbl 0192.36202