Given two weighted shift matrices \(A\) and \(B\), with nonzero weights \(a_1,\dots, a_{n-1}\) for \(A\), it is shown that \( B\) is unitarily equivalent to \(A\) if and only if \(b_1 \cdots b_n = a_1 \cdots a_n\) and, for some fixed \(k\), \(1 \leq k \leq n\), \(| b_j| = | a_{k+j}| (a_{n+j }\equiv a_j\)) for all \(j\). Next, it is shown that \(A\) is reducible if and only if \( {| a_j|}^n_{j=1}\) is periodic. Finally, it is proved that \(A\) and \(B\) have the same numerical range if and only if \(a_1 \cdots a_n = b_1 \cdots b_n\) and \(S_r (| a_1|^2,\dots , | a_n|^2) = S_r (| b_1|^2, \dots , | b_n|^2)\) for all \(1 \leq r \leq \lfloor n/2\rfloor\), where \(S_r\) is the circularly symmetric function.