Let $\mathbf{A}$ and $\mathbf{B}$ be $n\times n$ and $m\times n$ matrices, respectively, the pair $(\mathbf{A},\mathbf{B})$ is stabilizable if there exists an $n\times m$ matrix $\mathbf{K}$ such that $\rho(\mathbf{A}+\mathbf{B}\mathbf{K}')<1$, with $\rho(\cdot)$ denoting the spectral radius.
Stabilizability of $(\mathbf{A},\mathbf{B})$ can also be reached if for all eigenvalues $\lambda$ of $\mathbf{A}$ such that $|\lambda|\geq1$ holds that $\text{rank}[\lambda \mathbb{I} - \mathbf{A}, \mathbf{B}]=n$, with $\mathbb{I}$ the identity matrix.
Question: Can sufficient conditions on $\mathbf{A}$ and $\mathbf{B}$ be found such that the pair $(\mathbf{A},\mathbf{B})$ is stabilizable? For example, what if $m=n$ and $\rho(\mathbf{A})<1$?
I tried to find an upper bound for $\rho(\mathbf{A}+\mathbf{B}\mathbf{K}')$ and work with the ranks, but did't go anywhere. Any suggestion?
I am not sure of what the person who wrote the answer wants, so I will start with something and update my answer according to the following discussion, if there is any.
So, there is a necessary and sufficient condition for stabilizability that can be stated in terms of a Linear Matrix Inequality (LMI). Stabilizability of the pair $(A,B)$ is equivalent to the existence of a symmetric positive definite matrix $P$ (abbrv. $P\succ0$) such that
$$N_B(APA^T-P)N_B^T\prec0,$$ where $N_B$ is a basis of the left nullspace of $B$; i.e. $N_BB=0$ and $N_B$ of maximum rank.
This a condition in terms of $A$ and $B$, but it is implicit. When this condition holds, then there exists a $K$ such that $\rho(A+BK)<1$.
It can be generalized to $$N_B(APA^T-\alpha^{2}P)N_B^T\prec0$$ with $\alpha>0$. In this case there will exist a $K$ such that $\rho(A+BK)<\alpha$.
I am not sure how well this answers the question and I am happy to discuss about it.