On the Euclidean distance to the set of matrices with a multiple zero eigenvalue

Let $M_n(\mathbb C)$ be the set of $n$-by-$n$ complex matrices ($n>2$), and let $\mathcal K$ and $\mathcal L$ be the subsets of $M_n(\mathbb C)$ consisting of the matrices with a rank not greater than $n-2$ and of the matrices with a multiple zero eigenvalue, respectively. It is known that the minimal distance from a matrix $A\in M_n(\mathbb C)$ to the matrices in $\mathcal K$ is attained at the same matrix $K_A$ for both the spectral and Euclidean norm. It is shown that, for the set $\mathcal L$, similar minimal distances are attained, in the general case, at different matrices in $\mathcal L$. Moreover, the Euclidean distance from $A$ to $\mathcal L$ is, in general, strictly less than the Euclidean distance from $A$ to $\mathcal K$.