How these two statements equivalent?

Question

Given that $A$ is $n\times n$ matrix, $B$ is $n\times m$ matrix, could anyone make me understand how these two statements are equivalent?

$(1)$ rank $[A-\lambda I,B]=n\forall \lambda\in\text{ spec }A$

$(2)$ $\forall \lambda\in\text{ spec }A,\forall x\in\mathbb{C}^n $ we have $xA=\lambda x,xB=0\Rightarrow x=0$

Thanks for helping in details.

Answer 1

Given a complex $n\times(n+m)$ matrix $C$, these two statements are equivalent:

1. $\operatorname{rank}C=n$

2. $\forall x\in\Bbb{C}^n,~ xC=0\implies x=0$, that is, the nullity of $C^T$ is $0$

Proof. Observing that $\operatorname{rank}C^T=\operatorname{rank}C$, the proof comes from the rank-nullity theorem whereby $\operatorname{rank}C^T+\operatorname{nullity}C^T=n$.

The proof of your theorem comes from the fact that by $[C_1, C_2]$ with $C_1$ an $n\times n$ matrix and $C_2$ an $n\times m$ matrix it is meant a block partitioned $n\times (n+m)$ matrix whose blocks in order from left to right are $C_1$ and $C_2$.