Diagonals and numerical ranges of weighted shift matrices

For any n-by-n matrix A, we consider the maximum number k = k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of A. For any such compression, we give a standard model under unitary equivalence for A. This is then applied to determine the value of k(A) for A of size 3 in terms of the shape of W(A). When A is a matrix of the form0w10⋱⋱wn-1wn0,we show that k(A)=n if and only if either |w1|=⋯=|wn| or n is even and |w1|=|w3|=⋯=|wn-1| and |w2|=|w4|=⋯=|wn|. For such matrices A with exactly one of the wj's zero, we show that any k, 2⩽k⩽n-1, can be realized as the value of k(A), and determine exactly when the equality k(A)=n-1 holds.