Question

If the system$\frac{\mathbf{d}\overrightarrow{x}}{\mathrm{dt}}=A\overrightarrow{x}$is stable, is$\frac{\mathbf{d}\overrightarrow{x}}{\mathrm{dt}}\mathbf{=}{\mathbf{A}}^{-1}\overrightarrow{x}$stable as well? How can you tell?

Short answer

Thus, the system $\frac{d\overrightarrow{x}}{dt}=A^{-1}\overrightarrow{x}$will remain stable.

Step-by-step solution

Given in the question. 

The given systems are:

$\begin{array}{l}\frac{d\overrightarrow{x}}{dt}=A\overrightarrow{x}\\ \frac{d\overrightarrow{x}}{dt}=A^{-1}\overrightarrow{x}\end{array}$

Step 2:Stabilities of the systems. 

If the system$\frac{d\overrightarrow{x}}{dt}=A\overrightarrow{x}$is stable then the system$\frac{d\overrightarrow{x}}{dt}=A^{-1}\overrightarrow{x}$is also stable. Because the Eigen values of the second system is the inverse of the first.

So, the signs of the Eigen values not change.

Hence, the system$\frac{d\overrightarrow{x}}{dt}=A^{-1}\overrightarrow{x}$will remain stable.