Inequalities of singular values and unitarily invariant norms for sums and products of matrices

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In this work, we investigate inequalities of singular values and unitarily invariant norms for sums and products of matrices. First, we give an another more concise and clear proof to inequality obtained by Chen and Zhang [6, Theorem 5]. Then, we establish an inequality for singular values. In addition, we also give a singular values inequality for sums and products of matrices. As applications of this inequality, we present some unitarily invariant norms inequalities.