Abstract
In this note, some singular value inequalities on majorisation for positive semi-definite matrices are given. First, the weak log-majorisation of the quantity \( f(A)g(B)\pm f(B)g(A)\) for certain functions is obtained, where A, \(B\ge 0\). As a consequence, the weak log-majorisation of the quantity \(A^{p}B^{q}\pm B^{p}A^{q}\) for p, \(q>0\) is also obtained. Finally, different estimates for the singular values of the quantity \(A^{p}B^{q}\pm B^{p}A^{q}\) are deduced for A, \(B>0\) and \(0<p\le q\).
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Acknowledgements
The author is grateful to the anonymous referee and editors for their valuable comments and helpful suggestions, which led to an improved version of this manuscript. This work is supported by the National Natural Foundation of China (Grant No. 11161040).
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Communicated by Gadadhar Misra.
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Zhao, J. Some singular value inequalities on majorisation for positive semi-definite matrices. Indian J Pure Appl Math 54, 259–266 (2023). https://doi.org/10.1007/s13226-022-00249-2
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DOI: https://doi.org/10.1007/s13226-022-00249-2