Search for a tool
Jordan Normal Form Matrix

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 $M = S. J. \bar{S}$.

Results

Jordan Normal Form Matrix -

Tag(s) : Matrix

Share
dCode and you

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!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Jordan Normal Form Matrix tool, so feel free to write! Thank you !

# Jordan Normal Form Matrix

## Jordan Matrix Calculator

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 $M = S. J. \bar{S}$.

### How to calculate the Jordan Normal Form for a matrix?

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 $M = S . J . \bar{S}$

### How to calculate a power of a Jordan matrix?

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

## Source code

dCode retains ownership of the online 'Jordan Normal Form Matrix' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Jordan Normal Form Matrix download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!