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Zero-dilation Index of a Finite Matrix

For an $n$-by-$n$ complex matrix $A$, we define its zero-dilation index $d(A)$ as the largest size of a zero matrix which can be dilated to $A$. This is the same as the maximum $k$ ($\ge 1$) for which 0 is in the rank-$k$ numerical range of $A$. Using a result of Li and Sze, we show that if $d(A) > \lfloor 2n/3\rfloor$, then, under unitary similarity, $A$ has the zero matrix of size $3d(A)-2n$ as a direct summand. It complements the known fact that if $d(A)>\lfloor n/2\rfloor$, then 0 is an eigenvalue of $A$. We then use it to give a complete characterization of $n$-by-$n$ matrices $A$ with $d(A)=n-1$, namely, $A$ satisfies this condition if and only if it is unitarily similar to $B\oplus 0_{n-3}$, where $B$ is a 3-by-3 matrix whose numerical range $W(B)$ is an elliptic disc and whose eigenvalue other than the two foci of $\partial W(B)$ is 0. We also determine the value of $d(A)$ for any normal matrix and any weighted permutation matrix $A$.