Sparse hyperspectral unmixing is a popular technique used for earth observation. Nonconvex sparse prior is often considered and the linearized ADMM is practically used to solve the problems. This note provides a detailed analysis that supports a recent consequence and experimental results dedicated to this class of algorithms [1]. We prove the convergence of the linearized ADMM for separable reweighted sparse hyperspectral unmxing but the extended algorithm and the proposition also cover relaxed scenarios where the variables are non-separable.