Summary: We present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of \((N-1)\) or fewer plane rotations, where \(N\) is the dimension of the matrix. Both the algorithms of \textit{R. B. Bendel} and \textit{M. R. Mickey} [Commun. Stat., Simulation Comput. B7, 163--182 (1978; Zbl 0386.62011)] as well as of \textit{N. N. Chan} and \textit{K.-H. Li} [J. Math. Anal. Appl. 91, 562--566 (1983; Zbl 0529.15006)] algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as \(X^*X\), we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties.