||In this thesis, we study the numerical range of a 2-by-2 block matrix with zero diagonal block. We show that if B∈M_(k−1,k) (k ≥ 3) satisﬁes BB*=I_(k−1), then the numerical range of the 2-by-2 block matrix is the convex hull of two ellipses inscribed in the square [−1, 1] × [−1, 1]. On the other hand, we also show that if B ∈ M_k (k ≥ 3) satisﬁes|
∥B∥=1, then the numerical range of the 2-by-2 block matrix has 4 line segments on its boundary. Among other things, we consider the 2-by-2 block matrix A ∈ M_4, and we give a suﬃcient and necessary condition in terms of entries of B for numerical range of A being the convex hull of two ellipses.
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