I begin with terminology I use in the question. A real square matrix $A$ is
- negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
- $\ast$-negative-stable if for every eigenvalue $\lambda$ of $A$, either $\lambda = 0$ or ${\mathrm{Re}}(\lambda) < 0$;
- nonpositive-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) \leqslant 0$.
I made up the term '$\ast$-negative-stable' and I would welcome better and/or established terminology. For example, the Laplacian matrix of a nonnegatively weighted (directed or undirected) graph is $\ast$-negative-stable.
To put it broadly, I am looking for what is known about matrix operations that preserve the above stability properties.
Let $A$ be a real $n{\times}n$ matrix and let $u$, $v$, $w$ be real $n{\times}1$ vectors. Consider the real $n{\times}n$ matrices $D = \mathrm{diag}(u)$ and $B = vw^{\mathrm{T}}$, and the real number $\alpha = w^{\mathrm{T}}v$. I am particularly interested in what additional conditions on the matrix $A$ would make the following implications true. (I do not mean simultaneously true.) They concern preserving stability from $A$ to $AD$ for the first three and from $A$ to $A+B$ for the last three.
- ( $A$ is negative-stable and $u$ is positive ) $\Rightarrow$ ( $AD$ is negative-stable )
- ( $A$ is $\ast$-negative-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is $\ast$-negative-stable )
- ( $A$ is nonpositive-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is nonpositive-stable )
- ( $A$ is negative-stable and $\alpha < 0$ ) $\Rightarrow$ ( $A + B$ is negative-stable )
- ( $A$ is $\ast$-negative-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is $\ast$-negative-stable )
- ( $A$ is nonpositive-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is nonpositive-stable )
In implications 2 and 5 about $\ast$-negative-stability, it would be acceptable to assume that $A$ is similar to a Laplacian matrix (but Laplacian matrices should not be assumed to be symmetric). Would that be sufficient?
Addendum 1
Here is a way $\ast$-negative stability can be useful in studying negative/Hurwitz stability. Suppose I know that a matrix $A$ is similar to $C = \begin{pmatrix} O_{p{\times}p} & O_{p{\times}q} \\\ S & T \end{pmatrix}$, where $T$ is nonsingular. Then $T$ is negative-stable if and only if $A$ is $\ast$-negative-stable, a potentially useful observation if $A$ looks easier to work with than $T$. The notion of nonpositive stability can become useful in similar (but not identical) circumstances.
$A=\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$
is "*-negative-stable" and "nonpositive-stable", yet $e^{At}$ diverges. Usually a condition on the multiplicities of purely imaginary eigenvalue is assumed in addition. $\endgroup$ – Federico Poloni May 16 '12 at 7:39