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An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a recent work, t
Externí odkaz:
http://arxiv.org/abs/2412.10197
Autor:
Wassenaar, Tsjerk A.
In this work, we show how the eigenstructures of summands are related to that of the sum. In particular, we show that the sum of two positive semidefinite matrices can be written as the inner product of two block matrices $\mathbf{C} = \mathbf{A} + \
Externí odkaz:
http://arxiv.org/abs/2412.09207
Autor:
Binimelis, Miquel Albertí
Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While b
Externí odkaz:
http://arxiv.org/abs/2412.05595
Pole-swapping algorithms, generalizations of bulge-chasing algorithms, have been shown to be a viable alternative to the bulge-chasing QZ algorithm for solving the generalized eigenvalue problem for a matrix pencil A - {\lambda}B. It is natural to tr
Externí odkaz:
http://arxiv.org/abs/2411.17671
Autor:
Batra, Prashant
Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum root-modulus we
Externí odkaz:
http://arxiv.org/abs/2411.16385
Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the
Externí odkaz:
http://arxiv.org/abs/2411.15935
Autor:
Barrett, Wayne, Fallat, Shaun, Furst, Veronika, Nasserasr, Shahla, Rooney, Brendan, Tait, Michael
The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor -1$ have $q(
Externí odkaz:
http://arxiv.org/abs/2411.12917
Classification in the sense of similarity is an important issue. In this paper, we study similarity classification in Topological Data Analysis. We define a pseudometric $d_{S}^{(p)}$ to measure the distance between barcodes generated by persistent h
Externí odkaz:
http://arxiv.org/abs/2411.09960
Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic wavegu
Externí odkaz:
http://arxiv.org/abs/2411.09584
Autor:
Giller, Graham L.
This note discusses some of the aspects of a model for the covariance of equity returns based on a simple "isotropic" structure in which all pairwise correlations are taken to be the same value. The effect of the structure on feasible values for the
Externí odkaz:
http://arxiv.org/abs/2411.08864