Some new perturbation bounds of the positive (semi) definite polar factor and the (sub) unitary polar factor for the (generalized) polar decomposition under the general unitarily invariant norm and the spectral norm are presented. By applying our new bounds to the weighted cases, the known perturbation bounds for the weighted polar decomposition are improved. 2014 Elsevier Inc. All rights reser...

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh-Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues. In the present paper, we derive upper bounds for proximity of the Ritz valu...

This paper explores the concept of reparametrization invariant norm (RPI-norm) for C1-functions that vanish at −∞ and whose derivative has compact support, such as C1 c -functions. An RPI-norm is any norm invariant under composition with orientation-preserving diffeomorphisms. The L∞-norm and the total variation norm are well-known instances of RPI-norms. We prove the existence of an infinite f...

Let A − λB and C − λD be two normal matrix pencils or Hermitian matrix pencils or definite matrix pencils, and let AXD −BXC = S. Our main results in this part are estimates of a unitarily invariant norm of the solution X under some conditions on the spectra of two pencils. §

In their 1976 paper, Nagel and Rudin characterize the closed unitarily Möbius invariant spaces of continuous $$L^p$$ -functions on a sphere, for $$1\le p<\infty $$ . this paper we provide an analogous characterization weak*-closed $$L^\infty sphere. We also investigate algebras

We study global fluctuations for singular values of $M$-fold products several right-unitarily invariant $N\times N$ random matrix ensembles. As $N\to\infty$, we show the their height functions converge to an explicit Gaussian field, which is log-correlated $M$ fixed and has a white noise component $M\to\infty$ jointly with $N$. Our technique centers on multivariate Bessel generating these spect...