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URL: https://nakafa.com/en/subjects/ai-ds/linear-methods/jordan-normal-form
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Learn Jordan Normal Form and matrix triangularization: canonical decomposition for non-diagonalizable matrices with Jordan blocks and eigenvalues.

---

## Matrix Triangularization

Even though matrix $$A$$ cannot be diagonalized, the following theorem still holds.

Visible text: Even though matrix cannot be diagonalized, the following theorem still holds.

For matrix $$A \in \mathbb{K}^{n \times n}$$, if the characteristic polynomial of $$A$$ can be factored into linear factors, then $$A$$ is similar to an upper triangular matrix $$R \in \mathbb{K}^{n \times n}$$. The entries on the diagonal of this upper triangular matrix are all the eigenvalues of $$A$$.

Visible text: For matrix , if the characteristic polynomial of can be factored into linear factors, then is similar to an upper triangular matrix . The entries on the diagonal of this upper triangular matrix are all the eigenvalues of .

This similarity relationship is expressed as:

Component: MathContainer
Children:

```math
R = S^{-1} \cdot A \cdot S
```

where $$S \in \mathbb{K}^{n \times n}$$ is an invertible matrix.

Visible text: where is an invertible matrix.

This upper triangular form can be described more precisely through the Jordan Normal Form.

## Jordan Normal Form

Let $$A \in \mathbb{K}^{n \times n}$$ with characteristic polynomial:

Visible text: Let with characteristic polynomial:

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Children:

```math
\chi_A(t) = (\lambda_1 - t)^{r_1} \cdot \ldots \cdot (\lambda_k - t)^{r_k}
```

with distinct eigenvalues $$\lambda_1, \ldots, \lambda_k \in \mathbb{K}$$. Then there exists an invertible matrix $$S \in \mathbb{K}^{n \times n}$$ such that:

Visible text: with distinct eigenvalues . Then there exists an invertible matrix such that:

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Children:

```math
S^{-1} \cdot A \cdot S = \begin{pmatrix} \lambda_1 \cdot I_{r_1} + N_1 & & 0 \\ & \ddots & \\ 0 & & \lambda_k \cdot I_{r_k} + N_k \end{pmatrix}
```

This form shows how a matrix can be organized into simpler blocks, where each block is associated with one particular eigenvalue.

## Jordan Block Structure

For each $$i = 1, \ldots, k$$, the Jordan block $$\lambda_i \cdot I_{r_i} + N_i$$ has a very distinctive structure. Imagine it like an almost perfect staircase, where each step has the same value (namely the eigenvalue $$\lambda_i$$), but there are "connectors" in the form of the number $$1$$ at certain positions that make this structure unique.

Visible text: For each , the Jordan block has a very distinctive structure. Imagine it like an almost perfect staircase, where each step has the same value (namely the eigenvalue ), but there are "connectors" in the form of the number at certain positions that make this structure unique.

Component: MathContainer
Children:

```math
\lambda_i \cdot I_{r_i} + N_i = \begin{pmatrix} \lambda_i & 1 & & & \\ & \ddots & \ddots & & \\ & & \ddots & 1 & \\ & & & \lambda_i & 0 \\ & & & & \lambda_i & 1 \\ & & & & & \ddots & \ddots \\ & & & & & & \ddots & 1 \\ & & & & & & & \lambda_i & 0 \\ & & & & & & & & \ddots & \ddots \\ & & & & & & & & & \ddots & 0 \\ & & & & & & & & & & \lambda_i \end{pmatrix} \in \mathbb{K}^{r_i \times r_i}
```

In this structure, the eigenvalue $$\lambda_i$$ dominates the main diagonal, while the number $$1$$ appears at certain positions above the diagonal (called the superdiagonal). The positions of $$0$$ and $$1$$ on the superdiagonal determine how the Jordan block is divided into smaller sub-blocks. This structure provides complete information about how the linear transformation operates on the vector space associated with that eigenvalue.

Visible text: In this structure, the eigenvalue dominates the main diagonal, while the number appears at certain positions above the diagonal (called the superdiagonal). The positions of and on the superdiagonal determine how the Jordan block is divided into smaller sub-blocks. This structure provides complete information about how the linear transformation operates on the vector space associated with that eigenvalue.