Amplifying power of two matrix

Question

Studying dynamic systems I've stumbled upon a doubt. I have the initial response of two discrete systems: $x_A(k)=A^kx_0$ and $x_B(k)=B^kx_0$, with A and B square matrices. Now, $||A||<||B||$. I'm interested in comparing the sum of components of $x_A$ and $x_B$. I found pretty obvious (but probably wrong!) that $$\sum\limits_{i=1}^n {x_A}_i < \sum\limits_{i=1}^n {x_B}_i$$. Or at least I think that this is true for $k -> \infty$.

Assumptions: matrix norm 1 is used. A,B matrix are made of positive (or zero) scalars, x0 > 0.

But..is it true? Why?

Answer 1

It is not true. Take $$ A=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}1 & -1\\-1 & 1\end{bmatrix},\quad x_0=\begin{bmatrix}1\\0\end{bmatrix}. $$ Then $\|A\|=1<\|B\|=2$, but $\sum x_A=1>\sum x_B=0$ for all $k$.

Update: take $$ A=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}\frac12 & 1\\0 & \frac12\end{bmatrix},\quad x_0=\begin{bmatrix}1\\1\end{bmatrix}. $$ Then $\|A\|_1=1<\|B\|_1=1.5$, however, $\sum x_A=2$ for all $k$ while $x_B\to 0$ since $B$ is stable, so $\sum x_B\to 0<2$ when $k\to +\infty$. The norms are too crude measures. The asymptotics of $x_A$ and $x_B$ depends more on the spectral radii of $A$ and $B$. Moreover, the corresponding part of $x$ (the sum) must be observable.