Monotonic decay of trajectory components

Question

For the dynamical system with trajectory described by $x(t)=e^{-At}x_0$ with $A\in \mathbb{R}^{n\times n}$ symmetric positive-definite, I can find a change of basis $y=P^Tx$ such that all the components of $y$ decay monotonically to zero, where $P$ is the orthogonal matrix formed by the eigen-vectors of $A$. Is it possible to prove that all the individual components of $x=Py$ also decay monotonically to zero?

Answer 1

The claim is not true, $x$ does not converge to zero monotonically.

Consider $A = \begin{bmatrix} 2 & -1 \\ -1 & 2\end{bmatrix}$, which is symmetric and positive definite. The corresponding dynamics of $x$ is $$\begin{aligned} \dot{x}_1 &= -2x_1 + x_2, \\ \dot{x}_2 &= x_1 -2 x_2.\end{aligned}$$ Choose the initial condition $x_1(0)=1$, $x_2(0)=3$. Then $\dot{x}_1(0)=1$, and it does not converge to zero monotonically.