Commutator Inequalities Associated with the Polar Decomposition
نویسندگان
چکیده
Let A = UP be a polar decomposition of an n×n complex matrix A. Then for every unitarily invariant norm ||| · |||, it is shown that ||| |UP − PU |||| ≤ |||A∗A−AA∗||| ≤ ‖UP + PU‖ |||UP − PU |||, where ‖·‖ denotes the operator norm. This is a quantitative version of the wellknown result that A is normal if and only if UP = PU . Related inequalities involving self-commutators are also obtained.
منابع مشابه
Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin
Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$, let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$. Dewan et al proved that if $p(z)$ has all its zeros in $|z| leq k, (kleq 1),$ with $s$-fold zeros at the origin then for every $alphainmathbb{C}$ with $|alpha|geq k$, begin{align*} max_{|z|=...
متن کاملReliable location-allocation model for congested systems under disruptions using accelerated Benders decomposition
This paper aims to propose a reliable location-allocation model where facilities are subject to the risk of disruptions. Since service facilities are expected to satisfy random and heavy demands, we model the congested situations in the system within a queuing framework which handles two sources of uncertainty associated with demand and service. To insure the service quality, a minimum limit re...
متن کاملFinding the polar decomposition of a matrix by an efficient iterative method
Theobjective in this paper to study and present a new iterative method possessing high convergence order for calculating the polar decompostion of a matrix. To do this, it is shown that the new scheme is convergent and has high convergence. The analytical results are upheld via numerical simulations and comparisons.
متن کاملOn the polar derivative of a polynomial
For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
متن کاملProperties of matrices with numerical ranges in a sector
Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...
متن کامل