Editing Jordan normal form (section)

== Consequences ==

One can see that the Jordan normal form is essentially a classification result for square matrices, and as such several important results from linear algebra can be viewed as its consequences.

=== Spectral mapping theorem ===

Using the Jordan normal form, direct calculation gives a spectral mapping theorem for the [[functional calculus|polynomial functional calculus]]: Let ''A'' be an ''n'' × ''n'' matrix with eigenvalues ''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub>, then for any polynomial ''p'', ''p''(''A'') has eigenvalues ''p''(''λ''<sub>1</sub>), ..., ''p''(''λ''<sub>''n''</sub>).

=== Characteristic polynomial ===
The [[characteristic polynomial]] of {{math|''A''}} is <math>p_A(\lambda)=\det (\lambda I-A)</math>. [[Matrix similarity|Similar matrices]] have the same characteristic polynomial.
Therefore, <math display="inline">p_A(\lambda)=p_J(\lambda)=\prod_i (\lambda-\lambda_i)^{m_i}</math>,
where <math>\lambda_i</math> is the ''i''th root of <math display="inline">p_J</math> and <math>m_i</math> is its multiplicity, because this is clearly the characteristic polynomial of the Jordan form of ''A''.

=== Cayley–Hamilton theorem ===

The [[Cayley–Hamilton theorem]] asserts that every matrix ''A'' satisfies its characteristic equation: if {{math|''p''}} is the [[characteristic polynomial]] of {{math|''A''}}, then <math>p_A(A)=0</math>. This can be shown via direct calculation in the Jordan form, since if <math>\lambda_i</math> is an eigenvalue of multiplicity <math>m</math>,
then its Jordan block <math>J_i</math> clearly satisfies <math>(J_i-\lambda_i I)^{m_i}=0</math>.
As the diagonal blocks do not affect each other, the ''i''th diagonal block of <math>(A-\lambda_i I)^{m_i}</math> is <math>(J_i-\lambda_i I)^{m_i}</math>; hence <math display="inline">p_A(A)=\prod_i (A-\lambda_i I)^{m_i}=0</math>.

The Jordan form can be assumed to exist over a field extending the base field of the matrix, for instance over the [[splitting field]] of {{math|''p''}}; this field extension does not change the matrix {{math|''p''(''A'')}} in any way.

=== Minimal polynomial ===

The [[Minimal polynomial (linear algebra)|minimal polynomial]] P of a square matrix ''A'' is the unique [[monic polynomial]] of least degree, ''m'', such that ''P''(''A'') = 0. Alternatively, the set of polynomials that annihilate a given ''A'' form an ideal {{mvar|I}} in ''C''[''x''], the [[principal ideal domain]] of polynomials with complex coefficients. The monic element that generates {{mvar|I}} is precisely ''P''.

Let ''λ''<sub>1</sub>, ..., ''λ''<sub>''q''</sub> be the distinct eigenvalues of ''A'', and ''s''<sub>''i''</sub> be the size of the largest Jordan block corresponding to ''λ''<sub>''i''</sub>. It is clear from the Jordan normal form that the minimal polynomial of ''A'' has degree {{math|Σ}}''s''<sub>''i''</sub>.

While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of '''elementary divisors'''. The elementary divisors of a square matrix ''A'' are the characteristic polynomials of its Jordan blocks. The factors of the minimal polynomial ''m'' are the elementary divisors of the largest degree corresponding to distinct eigenvalues.

The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace. If all elementary divisors are linear, ''A'' is diagonalizable.

=== Invariant subspace decompositions ===

The Jordan form of a ''n'' × ''n'' matrix ''A'' is block diagonal, and therefore gives a decomposition of the ''n'' dimensional Euclidean space into invariant subspaces of ''A''. Every Jordan block ''J''<sub>''i''</sub> corresponds to an invariant subspace ''X''<sub>''i''</sub>. Symbolically, we put

:<math>\mathbb{C}^n = \bigoplus_{i = 1}^k X_i</math>

where each ''X''<sub>''i''</sub> is the span of the corresponding Jordan chain, and ''k'' is the number of Jordan chains.

One can also obtain a slightly different decomposition via the Jordan form. Given an eigenvalue ''λ''<sub>''i''</sub>, the size of its largest corresponding Jordan block ''s''<sub>''i''</sub> is called the '''index''' of ''λ''<sub>''i''</sub> and denoted by {{math|''v''(''λ''<sub>''i''</sub>)}}. (Therefore, the degree of the minimal polynomial is the sum of all indices.) Define a subspace ''Y''<sub>''i''</sub> by

:<math> Y_i = \ker(\lambda_i I - A)^{v(\lambda_i)}.</math>

This gives the decomposition

:<math>\mathbb{C}^n = \bigoplus_{i = 1}^l Y_i</math>

where {{mvar|l}} is the number of distinct eigenvalues of ''A''. Intuitively, we glob together the Jordan block invariant subspaces corresponding to the same eigenvalue. In the extreme case where ''A'' is a multiple of the identity matrix we have ''k'' = ''n'' and ''l'' = 1.

The projection onto ''Y<sub>i</sub>'' and along all the other ''Y<sub>j</sub>'' ( ''j'' ≠ ''i'' ) is called '''the spectral projection of ''A'' at {{math|v<sub>''i''</sub>}}''' and is usually denoted by '''''P''(''λ''<sub>''i''</sub> ; ''A'')'''. Spectral projections are mutually orthogonal in the sense that {{math|1=''P''(''λ''<sub>''i''</sub> ; ''A'') ''P''(v<sub>''j''</sub> ; ''A'') = 0}} if ''i'' ≠ ''j''. Also they commute with ''A'' and their sum is the identity matrix. Replacing every v<sub>''i''</sub> in the Jordan matrix ''J'' by one and zeroing all other entries gives {{math|''P''(v<sub>''i''</sub> ; ''J'')}}, moreover if ''U J U''<sup>−1</sup> is the similarity transformation such that ''A'' = ''U J U''<sup>−1</sup> then ''P''(''λ''<sub>''i''</sub> ; ''A'') = ''U P''(''λ''<sub>''i''</sub> ; ''J'') ''U''<sup>−1</sup>. They are not confined to finite dimensions. See below for their application to compact operators, and in [[holomorphic functional calculus]] for a more general discussion.

Comparing the two decompositions, notice that, in general, {{math|''l'' ≤ ''k''}}. When ''A'' is normal, the subspaces ''X''<sub>''i''</sub>'s in the first decomposition are one-dimensional and mutually orthogonal. This is the [[spectral theorem]] for normal operators. The second decomposition generalizes more easily for general compact operators on Banach spaces.

It might be of interest here to note some properties of the index, {{math|''ν''(''λ'')}}. More generally, for a complex number ''λ'', its index can be defined as the least non-negative integer {{math|''ν''(''λ'')}} such that

:<math>\ker(A-\lambda I)^{\nu(\lambda)} = \ker(A-\lambda I)^m, \; \forall m \geq \nu(\lambda) .</math>

So {{math|''ν''(v) > 0}} if and only if ''λ'' is an eigenvalue of ''A''. In the finite-dimensional case, {{math|''ν''(v) ≤}} the algebraic multiplicity of {{math|v}}.

===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.