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

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# 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 calculated 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 calculated a power of a Jordan matrix?

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

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