Jordan matrix

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

where is a square matrix of order of the form

. The matrix is called the Jordan block of order with eigen value . Every block is defined by an elementary divisor (cf. Elementary divisors, see [5]).

For an arbitrary square matrix over an algebraically closed field there always exists a square non-singular matrix over such that is a Jordan matrix (in other words, is similar over to a Jordan matrix). This assertion is valid under weaker restrictions on : For a matrix to be similar to a Jordan matrix it is necessary and sufficient that contains all roots of the minimum polynomial of . The matrix mentioned above is called a Jordan form (or Jordan normal form) of the matrix . 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 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 of Jordan blocks of order with eigen value in a Jordan form of a matrix is given by the formula

where is the unit matrix of the same order as , is the rank of the matrix , and is , 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
[2]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
[3]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)
[4]	S. Lang, "Algebra" , Addison-Wesley (1974)
[5]	A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)
[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)
[7]	R. Steinberg, "Classes of elements of semisimple algebraic groups" , Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 277–283