Tool to calculate the Jordan Normal Form of a Matrix (by Jordan reduction of a square matrix). The Jordan matrix is used in analysis, from a matrix M, the Jordan decomposition provides 2 matrices S and J such that $ M = S. J. \bar{S} $.

Jordan Normal Form Matrix - dCode

Tag(s) : Matrix

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Tool to calculate the Jordan Normal Form of a Matrix (by Jordan reduction of a square matrix). The Jordan matrix is used in analysis, from a matrix M, the Jordan decomposition provides 2 matrices S and J such that $ M = S. J. \bar{S} $.

Take $ M $ a square matrix of size $ n $, which has for eigen values the set of $ \lambda_n $.

__Example:__ $$ M = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 1 & 2 \end{bmatrix} \Rightarrow \lambda_n = \begin{pmatrix} 3 \\ 3 \\ 3 \end{pmatrix} $$

A matrix $ M $ of size $ n \times n $ is diagonalizable if and only if the sum of the dimensions of its eigen spaces is $ n $.

If $ M $ is not diagonalisable, there exists an almost diagonal matrix $ J $, called Jordan Normal Form, of the form $$ \begin{bmatrix} \lambda_i & 1 & \; & \; \\ \; & \lambda_i & \ddots & \; \\ \; & \; & \ddots & 1 \\ \; & \; & \; & \lambda_i \end{bmatrix} $$

__Example:__ Here, $ M $ has only 2 eigen vectors : $ v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} $ et $ v_2 = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} $, so is not diagonalizable, but has for **Jordan matrix** (canonical form) $$ M=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{bmatrix} $$

__Example:__ Alternative method: calculate the matrix $ S $ by finding a third vector $ v_3 $ such as $ (M - 3 I_3) v_3 = k_1 v_1 + k_2 v_2 \Rightarrow v_3 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} $. So $$ S = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} $$ and $ M = S . J . \bar{S} $

If $ M = SJS^{-1} $ Then $ M^k = SJ^kS^{-1} $ (see matrix powers).

dCode retains ownership of the source code of the script Jordan Normal Form Matrix online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Jordan Normal Form Matrix script for offline use on PC, iPhone or Android, ask for price quote on contact page !

jordan,matrix,reduction,decomposition,normal,form,canonical

Source : https://www.dcode.fr/matrix-jordan

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲