1. A $2\times2$ repeated-eigenvalue matrix
Use $A=\begin{bmatrix}\lambda&a\\0&\lambda\end{bmatrix}$. When $a\ne0$, there is only one eigenvector direction.
2. Jordan chain visualization
The vector $e_2$ maps to a multiple of $e_1$ under $A-\lambda I$, and $e_1$ maps to $0$.
3. Build a Jordan block $J_{\lambda,k}$
4. Matrix exponential
5. Kernel dimensions and block sizes
Enter $d_j=\dim\ker((A-\lambda I)^j)$. The differences $d_j-d_{j-1}$ count blocks of size at least $j$.
6. Minimal polynomial
Enter largest Jordan block size for each eigenvalue.
7. Independent-study practice tasks with answers
Task A. Decide whether $\begin{bmatrix}2&1\\0&2\end{bmatrix}$ is diagonalizable.
Answer: No. The eigenvalue $2$ has algebraic multiplicity $2$ but only a one-dimensional eigenspace.
Answer: No. The eigenvalue $2$ has algebraic multiplicity $2$ but only a one-dimensional eigenspace.
Task B. Find a Jordan chain for $\begin{bmatrix}2&1\\0&2\end{bmatrix}$.
Answer: $v_1=e_1$, $v_2=e_2$, because $(A-2I)v_1=0$ and $(A-2I)v_2=v_1$.
Answer: $v_1=e_1$, $v_2=e_2$, because $(A-2I)v_1=0$ and $(A-2I)v_2=v_1$.
Task C. Compute $J^m$ for $J=\begin{bmatrix}3&1\\0&3\end{bmatrix}$.
Answer: $J^m=\begin{bmatrix}3^m&m3^{m-1}\\0&3^m\end{bmatrix}$.
Answer: $J^m=\begin{bmatrix}3^m&m3^{m-1}\\0&3^m\end{bmatrix}$.
Similar practice. Compute $J^5$ for $J=\begin{bmatrix}2&1\\0&2\end{bmatrix}$.
Answer: $J^5=\begin{bmatrix}32&80\\0&32\end{bmatrix}$.
Answer: $J^5=\begin{bmatrix}32&80\\0&32\end{bmatrix}$.
Task D. If $d_1=3,d_2=5,d_3=6$, find block sizes.
Answer: Differences are $3,2,1$, so block sizes are $3,2,1$.
Answer: Differences are $3,2,1$, so block sizes are $3,2,1$.