Find all possible Jordan forms of a complex matrix with parameters

Question

Q: Given a matrix $$ A = \begin{pmatrix} a & b+c \\ b-c & -a \end{pmatrix}, $$ where $a,b,c \in \mathbb{C}$. Find all possible Jordan forms.

Answer 1

An easy calculation shows that the characteristic polynomial of $A$ is \begin{equation} X^2-a^2-b^2+c^2. \end{equation}

If $a^2+b^2-c^2\ne0$ then the eigenvalues $\pm\sqrt{a^2+b^2-c^2}$ are distinct. Hence $A$ is diagonalizable and all possible Jordan forms are \begin{equation} \begin{pmatrix}\pm\sqrt{a^2+b^2-c^2}&0\\0&\mp\sqrt{a^2+b^2-c^2}\end{pmatrix}. \end{equation}

Now assume $a^2+b^2-c^2=0$. Then $A$ has only one eigenvalue $0$. Moreover $A$ is diagonalizable if and only if $A=0$, i.e., $a=b=c=0$. Hence all possible Jordan forms of $A$ are \begin{align} &\begin{pmatrix} 0&0\\0&0 \end{pmatrix},&&\text{if $a=b=c=0$},\\ &\begin{pmatrix} 0&1\\0&0 \end{pmatrix},&&\text{otherwise}. \end{align}