Suppose $\Lambda$ is a diagonal matrix of size $n > 1$ and rank $1$, let's denote the sole element on the diagonal as $\lambda$. Consider the following equation: $\Lambda ^ {-1/2} = 1/\lambda * \Lambda ^{1/2}$ which is true, we can rewrite it as $(\Lambda ^ {-1/2} - 1/\lambda * \Lambda ^{1/2}) = 0 $.
Lets multiply it by $\Lambda^{1/2}$: $(\Lambda ^ {-1/2} - 1/\lambda * \Lambda ^{1/2}) * \Lambda^{1/2} = (I - 1/\lambda * \Lambda ) \neq 0$ (!) since $1/\lambda * \Lambda $ is a diagonal matrix with only one element equal to identity. I am pretty confused, because after multiplication of a zero matrix on a non zero one I have got non zero.
$\Lambda^{-1/2}$ is not defined.