# Nakafa Framework: LLM

URL: /en/subject/university/bachelor/ai-ds/linear-methods/jordan-normal-form
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/university/bachelor/ai-ds/linear-methods/jordan-normal-form/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Triangularization and Jordan Normal Form",
    description: "Master Jordan Normal Form and matrix triangularization: canonical decomposition for non-diagonalizable matrices with Jordan blocks and eigenvalues.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "07/17/2025",
    subject: "Linear Methods of AI",
};

## Matrix Triangularization

Even though matrix <InlineMath math="A" /> cannot be diagonalized, the following theorem still holds.

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

This similarity relationship is expressed as:

<MathContainer>
<BlockMath math="R = S^{-1} \cdot A \cdot S" />
</MathContainer>

where <InlineMath math="S \in \mathbb{K}^{n \times n}" /> is an invertible matrix.

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

## Jordan Normal Form

Let <InlineMath math="A \in \mathbb{K}^{n \times n}" /> with characteristic polynomial:

<MathContainer>
<BlockMath math="\chi_A(t) = (\lambda_1 - t)^{r_1} \cdot \ldots \cdot (\lambda_k - t)^{r_k}" />
</MathContainer>

with distinct eigenvalues <InlineMath math="\lambda_1, \ldots, \lambda_k \in \mathbb{K}" />. Then there exists an invertible matrix <InlineMath math="S \in \mathbb{K}^{n \times n}" /> such that:

<MathContainer>
<BlockMath 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}" />
</MathContainer>

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 <InlineMath math="i = 1, \ldots, k" />, the Jordan block <InlineMath math="\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 <InlineMath math="\lambda_i" />), but there are "connectors" in the form of the number 1 at certain positions that make this structure unique.

<MathContainer>
<BlockMath 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}" />
</MathContainer>

In this structure, the eigenvalue <InlineMath math="\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.