Prove that $H(x)=x^Tx$ is constant along solutions of the system

Question

Consider the diff. equation $\dot{x}=A(x)x$ where $A(x)$ is a matrix with real values of dimension n x n.

Prove that the function $H(x)=x^Tx$ is constant, along solutions of the system if $A(x)^T+A(x)=0$.

Also prove that the origin is Lyaponov stable fixed point if $A(x)^T+A(x)<0$

The function is constant if the derivative is set equal to zero. I have really no idea how to solve this and I would appreciate it a lot if someone could take a minute to show me how to prove this so that I can learn from it.

Answer 1

You have $$\dot{H}(x)=\dot{x^T}x+x^T \dot{x}=\dot{x}^T x +x^T \dot{x}=(A(x)x)^T x + x^T A(x)x = x^T (A(x)^T +A(x))x=0$$ proving that the function $H(x)$ is constant.

If $A(x)^T+A(x)<0$ is a negative matrix then for all $x$, $\dot{H}(x) < 0$, which means that the norm of $x$, namely $H(x)=x^Tx$ is decreasing and therefore that the origin is Lyaponov stable fixed point