Abstract
We obtain a characterization of pair matrices A and B of order n such that sjA≤sjB, j=1, …, n, where sjX denotes the j-th largest singular values of X. It can imply not only some well-known singular value inequalities for sums and direct sums of matrices but also Zhan’s result related to singular values of differences of positive semidefinite matrices. In addition, some related and new inequalities are also obtained.
Highlights
Let Mn denote the vector space of all complex n × n matrices, and let Hn be the set of all Hermitian matrices of order n
For A, B ∈ Hn, we use the notation A ≤ B or B ≥ A to mean that B − A is positive semidefinite
For T ∈ Mn, the singular values of T, denoted by s1(T), s2(T), . . . , sn(T), are the eigenvalues of the positive semidefinite matrix |T| (T ∗ T)1/2, enumerated as s1(T) ≥ s2(T) ≥ · · · ≥ sn(T) and repeated according to multiplicity. It follows that the singular values of a normal matrix are just the moduli of its eigenvalues
Summary
Let Mn denote the vector space of all complex n × n matrices, and let Hn be the set of all Hermitian matrices of order n. For T ∈ Mn, the singular values of T, denoted by s1(T), s2(T), . It follows that the singular values of a normal matrix are just the moduli of its eigenvalues. If T ∈ Mn is positive semidefinite, singular values and eigenvalues of T are the same. Recall that a complex matrix C ∈ Mn is called contraction if C∗C ≤ I, or equivalently ‖C‖ ≤ 1, where ‖ · ‖I denotes the spectral norm, the identity matrix of order n, respectively.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
