Miao's inequalities are a set of inequalities relating the image of the singular values of two matrices by a concave function. It was conjectured by W. Miao and proved in 2016.[1]
Statement
editLet f be a concave function f:R+→R+ with f(0)=0 and let X and Y be n×n complex matrices. The following inequality is valid:
where σ1(M) ≥ σ2(M) ≥ ... ≥ σn(M) denotes the singular values of the matrix M arranged in non-increasing order.
References
edit- ^ Yue, Man-Chung; So, Anthony Man-Cho. "A perturbation inequality for concave functions of singular values and its applications in low-rank matrix recovery". Applied and Computational Harmonic Analysis. 40 (2): 396–416. doi:10.1016/j.acha.2015.06.006.