Can the Kalman gain matrix contain negative values or absolute values greater than 1?
Sometimes you can answer a Kalman filtering question using lots of logic and very little math.  This is nice, because usually Kalman filtering is this big nasty blob of linear algebra and statistics all mashed together.

Take a 1-D system
$$ x_n = x_{n-1} + w_n, y_n = -0.01 x_n + v_n $$

At time \$n=0\$, let the "covariance" of \$x_0\$ be \$P = \infty\$, and let our pre-correction estimate of \$x_0 = 0\$.  Logically, we know absolutely nothing about \$x\$ until we measure \$y_0\$.  At that point, our logical, very best estimate of \$x_0\$ is \$-100 y_0\$.  Now if we reverse-engineer that statement, we find that \$K_0 = -100\$ for this problem, at this time.

You can do the math on this yourself, formally.  You'll probably need to take it in the limit as \$P \to \infty\$ to make the math work, but it'll work.  Trust me.   I'm an engineer; I know these things.