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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) to get, by decomposition, 2 matrices S and J such that M = S . J . S̄

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Jordan Normal Form Matrix -

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# Jordan Normal Form Matrix

## Jordan Matrix Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

### What is a Jordan matrix? (Definition)

A square 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 $J$, so-called Jordan matrix, which has nonzero elements on the main diagonal and on the first diagonal above. More precisely, the Jordan matrix will have the eigenvalues $\lambda_i$ on the diagonal and sometimes from 1 just above (in case of multiplicity), i.e. of the normal form of Jordan $$\begin{bmatrix} \lambda_i & 1 & \; & \; \\ \; & \lambda_i & \ddots & \; \\ \; & \; & \ddots & 1 \\ \; & \; & \; & \lambda_i \end{bmatrix}$$

### 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 $\lambda_i$.

Example: $$M = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 1 & 2 \end{bmatrix} \Rightarrow \lambda_1 = \lambda_2 = 3, \lambda_3 = 4$$ 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) $$J = \begin{bmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}$$

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 $$S = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$$ and $M = S . J . S^{-1}$

### What is Jordan Decomposition?

Jordan's decomposition is obtaining, from a matrix $M$, the matrices $S$ and $J$ such that $M = S . J . S^{-1}$

### What is Jordan reduction?

The reduction is the operation which makes it possible to pass from the matrix $M$ to the Jordan matrix $J$ (which is said to be reduced)

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

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

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Jordan Normal Form Matrix on dCode.fr [online website], retrieved on 2024-09-14, https://www.dcode.fr/matrix-jordan

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