Reduction of Jordan

See also: Jordan

The reduction of Jordan is the matric translation of the reduction of the endomorphisms introduced by Jordan. This reduction is employed so much, in particular analyzes of it for the resolution of differential equations or to determine the general term of recurring Suite S, that it is named sometimes “jordanisation of the endomorphisms”.

It consists in expressing the matrix of a Endomorphisme in reduced a base known as bases of Jordan. The reduction consists in determining a Décomposition of Dunford i.e. to find a endomorphism diagonalisable and a Endomorphisme nilpotent such as both commutate and that their amount is equal to the initial endomorphism, then on each space characteristic a reduction of Jordan on the factor the endomorphism nilpotent.

Construction of the base of Jordan

Either U a endomorphism on a vector Space E such as its minimal Polynomial or divided. It has the following properties then:

* E is the direct sum of the spaces characteristic of U . They are noted here E_i and the eigenvalues associated λ_i .

*La restriction of U on E_i is the sum of a homothety of report/ratio λ_i and of a endomorphism nilpotent noted n_i .

These results are shown in the article decomposition of Dunford.

* There exists a base E_ij of E_i $(e\_\; \{i1\},\; e\_\; \{i2\},\; \backslash \; cdots,\; e\_\; \{ip\_i\})\; \backslash ;$ such as $n\_i\; (e\_\; \{ij\})\; =k\_\; \{ij\}\; e\_\; \{ij+1\}\; \backslash ;$ where $k\_\; \{ij\}\; \backslash ;$ is equal either to 0 or with 1 and $n\_i\; (e\_\; \{ip\_i\})\; =0\; \backslash ;$.

This result is shown in the article Endomorphisme nilpotent.

Blocks of Jordan

One calls block of Jordan a matrix of the form

One calls block of Jordan nilpotent such a matrix where the diagonal coefficients all are null, i.e. form

Jordanisation of a endomorphism in a body algebraically closed

One considers a endomorphism in a vector space of finished size, of characteristic Polynôme divided. The theorem of Jordan informs us that he admits a matric representation of the following form

$$

\begin{bmatrix} \ mathcal {J} _ {\ lambda_1} & & & & \ \ & \ mathcal {J} _ {\ lambda_2} & & & \ \ & & \ ddots & & \ \ & & & \ ddots & \ \ & & & & \ mathcal {J} _ {\ lambda_r} \ \ \end{bmatrix} where the scalars $\backslash \; lambda\_i$ are the eigenvalues of the endomorphism considered.

Thus on a Body algebraically closed, and for example in $\backslash \; mathbb\; \{C\}$, any endomorphism admits a decomposition of this type.

Attention: there is not a priori a block of Jordan for each eigenvalue, several $\backslash \; lambda\_i$ can have the same value.

Properties of the blocks

Let us take a endomorphism U admitting such a representation. One studies a particular eigenvalue $\backslash \; lambda$ of the endomorphism U. One gathers together the vectors associated with the blocks $\backslash \; mathcal\; \{J\}\; \_\; \{\backslash \; lambda\}$. They form the characteristic space associated with the eigenvalue $\backslash \; lambda$. It is a stable space on which $u-\; \backslash \; lambda\; Id$ induces a Endomorphisme nilpotent $n\_\; \backslash \; lambda$.

• the multiplicity of $\backslash \; lambda$ (multiplicity in the characteristic Polynomial) is equal to the dimension of characteristic space.
• the multiplicity of $\backslash \; lambda$ in the minimal Polynôme is equal to the index of nilpotence of the endomorphism $n\_\; \backslash \; lambda$.

Application to the classes of similarity of the matrices

One places oneself on a body algebraically closed. Two matrices are similar if and only if they have the same writing in blocks of Jordan, to the order nearly the blocks.

Examples of reduction of Jordan

Let us examine the methods of determination of the matrices of passage for two examples.

Example 1

Let us determine the matrix of passage for the following example:

$A=\backslash begin\{pmatrix\}$

322 & -323 & -323 & 322 \ \ 325 & -326 & -325 & 326 \ \ -259 & 261 & 261 & -260 \ \ -237 & 237 & 238 & -237 \ end {pmatrix}.

Let us seek the characteristic spaces, i.e. vectors X solutions of

$(\backslash \; lambda\; I\; Has)\; ^k\; \backslash \; mathbf\; \{X\}\; =\; \backslash \; mathbf\; \{0\}$

Who will allow us to determine the vector series whose elements form the columns of the matrix of passage.

Let us notice whereas 5 is eigenvalue and that the first vector of the basis of definition of the matrix has for associated minimal polynomial (X-5) ⁴ . Its characteristic space is thus whole space. If we then note v this vector, the family made up of the elements ( has   −  5 I ) ³ ( v ), ( has   −  5 I ) ² ( v ), ( has   −  5 I ) ( v ) and v form a base of Jordan.

$\backslash \; left\; \backslash \; \{(A-5I)\; ^3\; \backslash \; mathbf\; \{v\},\; (A-5I)\; ^2\; \backslash \; mathbf\; \{v\},\; (A-5I)\; \backslash \; mathbf\; \{v\},\; \backslash \; mathbf\; \{v\}\; \backslash \; right\; \backslash \}$

$=\; \backslash \; left\; \backslash \; \{$

\begin{pmatrix} 5922 \\ 4230 \\ -3572 \\ -5170 \end{pmatrix}, \begin{pmatrix} 2857 \\ 2363 \\ -1962 \\ -2392 \end{pmatrix}, \begin{pmatrix} 317 \\ 325 \\ -259 \\ -237 \end{pmatrix}, \ begin {pmatrix} 1 \ \ 0 \ \ 0 \ \ 0 \ end {pmatrix} \ right \}

We determined the matrix of passage:

$P=\backslash begin\{pmatrix\}$

5922 & 2857 & 317 & 1 \ \ 4230 & 2363 & 325 & 0 \ \ -3572 & -1962 & -259 & 0 \ \ -5170 & -2392 & -237 & 0 \ \ \end{pmatrix}.

And stamps it of Jordan is the following one:

$J=J\_4\; (5)\; =\; \backslash \; begin\; \{pmatrix\}$

5 & 1 & 0 & 0 \ \ 0 & 5 & 1 & 0 \ \ 0 & 0 & 5 & 1 \ \ 0 & 0 & 0 & 5 \ end {pmatrix}.

Example 2

Let us consider the following example

$B\; =$

\begin{pmatrix} 5 & 4 & 2 & 1 \ \ 0 & 1 & -1 & -1 \ \ -1 & -1 & 3 & 0 \ \ 1 & 1 & -1 & 2 \ \ \end{pmatrix}.

The eigenvalues of B are 4,4,2 and 1. Moreover, one notices that:

$\backslash \; mathrm\; \{dim\}\; \backslash \; \backslash \; ker\; \{(BI)\}\; =\; 1,\; \backslash \; mathrm\; \{dim\}\; \backslash \; \backslash \; ker\; \{(B-2I)\}\; =\; 1,$

\ mathrm {dim} \ \ ker {(B-4I)} = 1, \ mathrm {dim} \ \ ker ({B-4I}) ^2 = 2

We deduce from it that the vector space breaks up into direct Somme following:

$J=J\_1\; (1)\; \backslash \; oplus\; J\_1\; (2)\; \backslash \; oplus\; J\_2\; (4)$

We notice that the vector column (0,0, −1,1) ^T has as an image by the matrix (- 1,0,1, - 1) ^T . These two vectors columns generate space characteristic of eigenvalue 4.

One from of deduced

$\backslash \; ker\; \{(B-4I)\}^2\; =\; \backslash \; mathrm\; \{Gen\}\; \backslash \; \backslash \; left\; \backslash \; \{\backslash \; begin\; \{pmatrix\}\; 0\; \backslash \; \backslash \; 0\; \backslash \; \backslash \; -1\; \backslash \; \backslash \; 1\; \backslash \; end\; \{pmatrix\},\; \backslash \; begin\; \{pmatrix\}\; -1\; \backslash \; \backslash \; 0\; \backslash \; \backslash \; 1\; \backslash \; \backslash \; -1\; \backslash \; end\; \{pmatrix\}\; \backslash \; right\; \backslash \}$

We deduce from it the matrix from passage and the form of Jordan:

$P=\backslash begin\{pmatrix\}$

-1 & 0 & 1 & -1 \ \ 0 & 0 & -1 & 1 \ \ 1 & -1 & 0 & 0 \ \ -1 & 1 & 1 & 0 \ end {pmatrix} = \ left ((B-4I) \ mathbf {v} \ left|\ mathbf {v} \ left|\ mathbf {W} \ left|\ mathbf {X} \ right) \ right. \ right. \ right.

and

$P^\; \{-\; 1\}\; BP=J=\; \backslash \; begin\; \{pmatrix\}$

4 & 1 & 0 & 0 \ \ 0 & 4 & 0 & 0 \ \ 0 & 0 & 2 & 0 \ \ 0 & 0 & 0 & 1 \ end {pmatrix}.

Reduction of Jordan and differential connections

A system of linear differential equations in $y$ can be reduced to a matric differential equation of order 1: $u\text{'}\; (T)\; =\; With\; (T)\; the$ and the initial condition $U\; (0)\; =\; u\_0$, where $u\; (T)$ is a vector column containing the dériviées successive ones of $y$. The resolution is then explicit: $U\; (T)\; =\; \backslash \; exp\; (your)\; u\_0$. The advantage of the normal form of Jordan lies in the facility of calculations of the matrices of the blocks of Jordan. Indeed, the exponential one of a block of Jordan nilpotent of size $p$ is

One sees this manner the calculative interest of this method.

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