16,258 research outputs found
Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms
We improve and expand in two directions the theory of norms on complex
matrices induced by random vectors. We first provide a simple proof of the
classification of weakly unitarily invariant norms on the Hermitian matrices.
We use this to extend the main theorem in [7] from exponent to . Our proofs are much simpler than the originals: they do not require
Lewis' framework for group invariance in convex matrix analysis. This
clarification puts the entire theory on simpler foundations while extending its
range of applicability.Comment: 10 page
Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem
In this paper, we introduce properly-invariant diagonality measures of
Hermitian positive-definite matrices. These diagonality measures are defined as
distances or divergences between a given positive-definite matrix and its
diagonal part. We then give closed-form expressions of these diagonality
measures and discuss their invariance properties. The diagonality measure based
on the log-determinant -divergence is general enough as it includes a
diagonality criterion used by the signal processing community as a special
case. These diagonality measures are then used to formulate minimization
problems for finding the approximate joint diagonalizer of a given set of
Hermitian positive-definite matrices. Numerical computations based on a
modified Newton method are presented and commented
The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach
In contemporary applied and computational mathematics, a frequent challenge
is to bound the expectation of the spectral norm of a sum of independent random
matrices. This quantity is controlled by the norm of the expected square of the
random matrix and the expectation of the maximum squared norm achieved by one
of the summands; there is also a weak dependence on the dimension of the random
matrix. The purpose of this paper is to give a complete, elementary proof of
this important, but underappreciated, inequality.Comment: 20 page
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