Zero-trace index of matrices

The zero-trace index $t(A)$ of a square complex matrix $A$ is introduced. It is defined as the largest integer $k$ such that $A$ admits a $k\times k$ compression with trace zero. This concept is analogous to the zero-dilation index $d(A)$ and is closely related to the $k$-numerical range. The $k$-numerical range is characterized as an intersection of half-planes, which is used to derive a formula for $t(A)$ that reduces its computation to the Hermitian case. Conditions for $t(A)=d(A)$ are also established, and matrices satisfying $t(A)=n-1$ are characterized. Finally, the zero-trace index is determined for several classes of matrices.