在本篇論文中，對任意3×3的複數矩陣A和B，我們給出了充分且必要的條件對於AB矩陣乘積的數值域和BA矩陣乘積的數值域相等時。此外，去研究當A和A2的數值域半徑為1且A3的數值域半徑小於1時，A會有什麼樣的矩陣結構。以及最後，我們給出了充分且必要的條件對於當A為壓縮矩陣其特徵值長度皆小於1且A的範數為1，A與B張量積的數值域半徑等於A的範數與B的數值域半徑乘積時。;In this thesis, for any two 3-by-3 complex matrices A and B, we show that the necessary and suffi cient conditions for the equality W(AB) = W(BA) to hold, where W( ) denotes the numerical range of a matrix, a nd the structure of A when w(A) =w (A2) = 1 and w (A3) < 1, where w( ) denotes the numerical radius of a matrix, and obtain the necessary and suffi cient condition for the equality w(A B) = kAkw(B)to hold when A is a completely nonunitary contraction with kAk = 1, where k k denotes the usual operator norm of a matrix.
