I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and the properties of a specific block of the blockmatrix A:
Given the matrix pair $$ (E,A)= \left( \begin{bmatrix}I_r&0\\\ 0&0\end{bmatrix},\begin{bmatrix}A_{11}&A_{21}\\\ A_{12}&A_{22}\end{bmatrix} \right) $$
where $E,A \in C^{m \times n}$ and $I_r$ is the identity matrix of dimension $r < \min (m,n)$ one have to proof that the given matrix pair is regular and of index one if and only if $A_{22}$ is a square block and nonsingular.
I have some ideas, but don´t know how to proceed from there. For the $ \Rightarrow $ direction I know, that from regularity of $(E,A)$ it follows:
- $E,A$ must be square matrices
- $(E,A)$ is equivalent to the canonical form $ \left( \begin{bmatrix}I&0\\\ 0&N\end{bmatrix},\begin{bmatrix}J&0\\\ 0&I\end{bmatrix} \right) $ , i.e. there are nonsingular Matrices $P,Q$ such that $PEQ = \begin{bmatrix}I&0\\\ 0&N\end{bmatrix}$ and $PAQ = \begin{bmatrix}J&0\\\ 0&I\end{bmatrix}$ where $J$ is a matrix in jordan normal form and $N$ is nilpotent in Jordan normal form
- $p(s) = \det( sE-A )$ is not $0$ for every $s$
Also, because the index of $(E,A)$ is $\nu=1$ it is that $N^{0} \neq 0,\;N^{1}=0$ the nilpotent block should be of size $1\times1$ : $$N = [0]$$ since $N^0=[0]^0=1$ and $N^1=[0]$. But from here I dont know how to continue. Thank you in advance.