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Section 8.1 Jordan Blocks and Jordan Canonical Form
Definition 8.1.1 . Jordan Block.
A Jordan block of size \(m \times m\) with eigenvalue \(\lambda\) is a matrix of the form:
\begin{equation*}
J_m(\lambda) = \begin{pmatrix}
\lambda \amp 1 \amp 0 \amp \cdots \amp 0 \amp 0 \\
0 \amp \lambda \amp 1 \amp \cdots \amp 0 \amp 0 \\
\vdots \amp \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots \\
0 \amp 0 \amp 0 \amp \cdots \amp \lambda \amp 1 \\
0 \amp 0 \amp 0 \amp \cdots \amp 0 \amp \lambda
\end{pmatrix}
\end{equation*}
A Jordan block has
\(\lambda\) on the main diagonal, 1’s on the superdiagonal (the diagonal immediately above the main diagonal), and 0’s elsewhere.
Example 8.1.2 . Examples of Jordan Blocks.
Here are some examples of Jordan blocks:
\begin{align*}
J_3(-1) \amp= \begin{pmatrix} -1 \amp 1 \amp 0 \\ 0 \amp -1 \amp 1 \\ 0 \amp 0 \amp -1 \end{pmatrix}\\
J_1(3) \amp= \begin{pmatrix} 3 \end{pmatrix}\\
J_2(2) \amp= \begin{pmatrix} 2 \amp 1 \\ 0 \amp 2 \end{pmatrix}
\end{align*}
Checkpoint 8.1.3 .
Find the characteristic polynomial of the Jordan block
\(J_3(-1)\) from the previous example.
Definition 8.1.4 . Jordan Canonical Form.
Theorem 8.1.5 . Existence of Jordan Canonical Form.
Every square matrix over
\(\mathbb{C}\) is similar to a matrix in Jordan canonical form. Moreover, this Jordan canonical form is unique up to the order of the Jordan blocks.
