Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1601.02586

Exact Relation between Singular Value and Eigenvalue Statistics

Mario Kieburg·Holger Kösters
View original source ↗Read source document ↗Export metadata ↓

Abstract

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

Explore related subjects

math.CAmath-phmath.MPmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mario Kieburg, Holger Kösters. 2016-02-05. Exact Relation between Singular Value and Eigenvalue Statistics. https://doi.org/10.1142/s2010326316500155

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spectrality of Weighted Measures on Two Line Segments

We study the spectrality of measures with positive integrable densities supported on two line segments in $\mathbb R^d$. We prove that, when the two segments are non-overlapping, spectrality forces the density on each segment to be constant almost everywhere. When the two segments are overlapping, spectrality forces the total density to be constant almost everywhere on their union. We then study the resulting measures with positive constant densities according to the geometric of the segments. For two non-coplanar segments, every such measure admits a spectrum contained in a straight line. For two segments lying on distinct parallel lines, the measure is spectral if and only if the two densities are equal. For non-parallel coplanar segments, a suitable invertible linear transformation converts the measure into an unweighted arc-length measure. When these segments are viewed in their affine plane, every spectrum of a spectral measure is contained in a straight line. Finally, we give examples showing how the densities determine the directions of line spectra and construct explicit spectra for weighted measures.

math.CA↗
arXiv2026

Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

math.CA↗
arXiv2026

Sparse bounds and joint oscillation for bilinear rough commutators

We study joint oscillation conditions for commutators of bilinear rough singular integrals with bounded mean-zero angular kernels. For second-order commutators with two possibly distinct complex-valued symbols, we prove lower bounds under a geometric non-degeneracy assumption. These bounds distinguish the commutation positions: mixed commutators control the product of the separate quadratic oscillations, while repeated-position commutators of the partial transposes additionally control the quadratic average of the product of the centered symbols. We also establish sparse domination for iterated commutators, retaining the symbol oscillations inside the local averages. For each fixed commutator order, the sparse constant is $O(s/(s-1))$ as the averaging exponent $s$ decreases to $1$. The resulting joint oscillation conditions yield strong bounds in the Banach range and weak bounds in the quasi-Banach range. At the natural exponent triples, the necessary and sufficient conditions differ by an arbitrarily small increase in the oscillation exponents.

math.CA↗