View source for Jordan matrix

A square block-diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543401.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543402.png" /> of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543404.png" /> is a square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543405.png" /> of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543407.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543408.png" /> is called the Jordan block of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434010.png" /> with eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434011.png" />. Every block is defined by an elementary divisor (cf. [[Elementary divisors|Elementary divisors]], see [[#References|[5]]]).

For an arbitrary square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434012.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434013.png" /> there always exists a square non-singular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434016.png" /> is a Jordan matrix (in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434017.png" /> is similar over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434018.png" /> to a Jordan matrix). This assertion is valid under weaker restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434019.png" />: For a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434020.png" /> to be similar to a Jordan matrix it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434021.png" /> contains all roots of the minimum polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434022.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434023.png" /> mentioned above is called a Jordan form (or Jordan normal form) of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434024.png" />. C. Jordan [[#References|[1]]] was one of the first to consider such a normal form (see also the historical survey in Chapts. VI and VII of [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434025.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434026.png" /> of Jordan blocks of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434027.png" /> with eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434028.png" /> in a Jordan form of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434029.png" /> is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434031.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434032.png" /> is the unit matrix of the same order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434035.png" /> is the [[Rank|rank]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434038.png" />, 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 [[#References|[2]]]–[[#References|[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 [[#References|[6]]]–[[#References|[7]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Jordan, "Traité des substitutions et des équations algébriques" , Paris (1870) pp. 114–125 {{MR|1188877}} {{MR|0091260}} {{ZBL|03.0042.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|0643362}} {{ZBL|1139.12001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian) {{MR|1657129}} {{MR|0107649}} {{MR|0107648}} {{ZBL|0927.15002}} {{ZBL|0927.15001}} {{ZBL|0085.01001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) {{MR|}} {{ZBL|0396.15001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|0192.36201}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Steinberg, "Classes of elements of semisimple algebraic groups" , ''Internat. Congress Mathematicians (Moscow, 1966)'' , Mir (1968) pp. 277–283 {{MR|0238856}} {{ZBL|0192.36202}} </TD></TR></table>