Editing Jordan normal form (section)

===Plane (flat) normal form===

The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space.

Sets of representatives of matrix conjugacy classes for Jordan normal form or [[rational canonical form]]s in general do not constitute linear or 
affine subspaces in the ambient matrix spaces.

[[Vladimir Arnold]] posed<ref>{{citation
 | last = Arnold | first = Vladimir I. | editor-first1 = Vladimir I. | editor-last1 = Arnold | author-link = Vladimir Arnold
 | contribution = 1998-25
 | doi = 10.1007/b138219
 | isbn = 3-540-20614-0
 | mr = 2078115
 | page = 127
 | publisher = Springer-Verlag | location = Berlin
 | title = Arnold's Problems
 | title-link = Arnold's Problems | year = 2004}}. See also comment, p. 613.</ref> a problem:
Find a canonical form of matrices over a field for which the set of representatives of matrix conjugacy classes is a union of affine linear subspaces (flats). In other words, map the set of matrix conjugacy classes injectively back into the initial set of matrices so that the image of this embedding—the set of all normal matrices, has the lowest possible degree—it is a union of shifted linear subspaces.

It was solved for algebraically closed fields by Peteris Daugulis.<ref name="originalpaper">{{cite journal | author = Peteris Daugulis |date=2012 | title = A parametrization of matrix conjugacy orbit sets as unions of affine planes| 
pages = 709–721 | journal = Linear Algebra and Its Applications | volume = 436 | issue = 3 |  
doi = 10.1016/j.laa.2011.07.032 |arxiv = 1110.0907 |s2cid=119649768 }}</ref> 
The construction of a uniquely defined '''plane normal form''' of a matrix starts by considering its Jordan normal form.