We discuss a matrix decomposition,  show the uniqueness and construction (of the $Z$ matrix in our main result) of the matrix decomposition, and give an affirmative answer to a question proposed in [J. Math. Anal. Appl. 407 (2013) 436-442]. The theorem is stated as Sectoral Decomposition:

 Let $A$ be an $n/times n$ complex matrix such that its numerical range is contained in a sector in the 1st and 4th quadrants, i.e., $W(A)/subseteq S_{/alpha}$ for some $/alpha /in [0, /frac{/pi}{2})$. Then there exist an invertible matrix $X$ and a unitary diagonal matrix $Z={/rm diag} (e^{i/theta_1}, /dots, e^{i/theta_n})$ with all $|/theta_j|/leq /alpha$  such that $A=XZX^*$. Moreover, such a matrix $Z$ is unique up to permutation.