Abstract: 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 here 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.