本論文探討對角方塊都是零的2×2方塊矩陣的數值域。我們證明當B是k×k(k>2)矩陣滿足B*B是k-1維的單位矩陣和一維0矩陣的直和，則此2×2方塊矩陣其數值域會是兩個內切在[-1,1]×[-1,1]正方形裡的橢圓的凸包。另一方面，只要B滿足∥B∥=1，我們也對此2×2方塊矩陣其數值域的邊界給出刻劃。此外，對於4階的2×2方塊矩陣 ，我們也給出其數值域會是兩個內切在[-1,1]×[-1,1]正方形裡橢圓的凸包的充分必要條件。

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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