I'm solving a homework question that asks me to do the following:
"List the five upper Jordan canonical forms for a $4\times 4$ matrix $A$ with a real eigenvalue $\lambda$ of multiplicity $4$ and give the corresponding deficiency indices in each case."
I can't seem to understand what they mean by "$5$ upper Jordan canonical forms"? Isn't the answer unique and straightforward with the canonical form:
$$ J_= \left[ {\begin{array}{cccc} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 &\lambda & 1 \\ 0 & 0 & 0 & \lambda \end{array} } \right] $$
And what do they mean by deficiency indices? Are those the "$1$'s" that appear on top of each $\lambda$?
The other four Jordan canonical forms are $$ \begin{bmatrix} \lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda \end{bmatrix} $$ The deficiency indices of a matrix $A\in\mathcal{M}_{n\times n}(\mathbb{C})$ are the numbers $$ n_{\pm}(A)=\dim\ker(A^*\mp i\cdot I_n) $$ Since $\lambda$ is assumed to be real, computing these numbers should be straight forward.