Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)$k=k(A)$ for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)$\partial W(A)$ of the numerical range W(A)$W(A)$ of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)$k(A)$ can be computed. For a matrix A   of the form B⊕C$B\oplus C$, we show that k(A)=2$k(A)=2$ if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2$k(C)=2$. For an irreducible matrix A  , we can determine exactly when the value of k(A)$k(A)$ equals the size of A  . These are then applied to determine k(A)$k(A)$ for a reducible matrix A   of size 4 in terms of the shape of W(A)$W(A)$.