Why does the negative reward function in LQR encourage convergence to the origin?

Question

I was reading Stanford's CS 229 materials on Linear Quadratic Regulation (LQR) (Lecture note 13, YouTube Lecture 18, around minute 36), and it mentions that:

[...] the quadratic formulation of the reward is equivalent to saying that we want our state to be close to the origin. For example, if $U_t = I_n $(the identity matrix) and $W_t = I_d$, then $R_t = -\Vert(s_t)\Vert^2-\Vert(a_t)\Vert^2$, meaning that we want to take smooth actions (small norm of $a_t$) to go back to the origin (small norm of $s_t$).

Why is that? Why the norm of the origin is smaller? $s_t$ for any time point are just a vector of n dimension which can take any value, right?

Answer 1

Minimizing $s^TUs=s^Ts=\|s\|^2$ minimizes the distance from the origin, because the norm of a vector is its Euclidean distance from the origin:

$$\|s\|= \sqrt{s_1^2+s_2^2+\dots+s_n^2} = \sqrt{(s_1-0)^2+(s_2-0)^2+\dots+(s_n-0)^2}$$