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Yorick Hardy

Publications and source records attributed to Yorick Hardy.

At least 19 recordsLinked to original sources
arXiv2026

Kronecker differences

Over the real numbers, the Kronecker sum is the unique operation on matrices which exponentiates to the Kronecker product. Kronecker quotients provide an algebraic view of decompositions of matrices in terms of Kronecker products. This article explores families of operations, Kronecker differences, which are a kind of "inverse" for Kronecker sums. The correspondence between Kronecker differences and Kronecker quotients is explored. Furthermore, we show that a certain class of Kronecker differences may be characterized by families of matrices with these families again being expressed as Kronecker products. This approach provides a different "nonlinear" view towards tensor decomposition.

math.RA↗
arXiv2026

Canonical forms for boundary conditions of self-adjoint odd-order differential operators

It is useful to have canonical forms of boundary conditions in the study of the eigenvalues of boundary value problems and associated numerical applications. In [J. Appl. Anal. Comput., 2024, 14(4), {1854--1868}], a canonical form is given for self-adjoint differential operators of even order, with eigenvalue parameter dependent boundary conditions. In this article, we derive canonical forms for the remaining case, namely: for self-adjoint $(2n+1)$-th order differential operators with eigenvalue parameter dependent boundary conditions.

math.SP↗
arXiv2021

Canonical forms for boundary conditions of self-adjoint differential operators

Canonical forms of boundary conditions are important in the study of the eigenvalues of boundary conditions and their numerical computations. The known canonical forms for self-adjoint differential operators, with eigenvalue parameter dependent boundary conditions, are limited to 4-th order differential operators. We derive canonical forms for self-adjoint 2n-th order differential operators with eigenvalue parameter dependent boundary conditions. We compare the 4-th order canonical forms to the canonical forms derived in this article.

math.CA↗
arXiv2019

Preserving the trace of the Kronecker sum

The aim of this paper is to study linear preservers of the trace of Kronecker sums and their connection with preservers of determinants of Kronecker products. The partial trace and partial determinant play a fundamental role in characterizing the preservers of the trace of Kronecker sums and preservers of the determinant of Kronecker products respectively.

math.RA↗
arXiv2018

Hierarchy of Spin Operators, Quantum Gates, Entanglement, Tensor Product and Eigenvalues

{\bf Abstract.} We show that two hierarchies of spin Hamilton operators admit the same spectrum. Both Hamilton operators play a central role for quantum gates in particular for the case spin-$\frac12$ and the case spin-1. The spin-$\frac12$, spin-1, spin-$\frac32$ and spin-2 cases are studied in detail. Entanglement and mutually unbiased bases of the eigenvectors is discussed. Two triple Hamilton operators are also investigated. Both are also admitting the same spectrum.

quant-ph↗
arXiv2018

Partial determinants of Kronecker products

Let $\det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $\det(\det_2(A)) = \det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary field. Further insisting that $\det_2(AB)=\det_2(A)\det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $\text{Det}$ which satisfies $\text{Det}(\text{Det}_2(A)) = \text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $\text{Det}$ is defined.

math.RA↗
arXiv2017

Matrices, Fermi Operators and Applications

We consider the vector space of $n \times n$ matrices over $\mathbb C$, Fermi operators and operators constructed from these matrices and Fermi operators. The properties of these operators are studied with respect to the underlying matrices. The commutators, anticommutators, and the eigenvalue problem of such operators are also discussed. Other matrix functions such as the exponential functions are studied. Density operators and Kraus operators are also discussed.

quant-ph↗
arXiv2016

A Short Note on Kronecker Square Roots

The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the characterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.

math.RA↗
arXiv2015

Boolean Functions, Quantum Gates, Hamilton Operators, Spin Systems and Computer Algebra

We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.

cs.MS↗
arXiv2014

Exponential of a Matrix, a Nonlinear Problem and Quantum Gates

We describe solutions of the matrix equation $\exp(z(A-I_n))=A$, where $z \in {\mathbb C}$. Applications in quantum computing are given. Both normal and nonnormal matrices are studied. For normal matrices, the Lambert W-function plays a central role.

math-ph↗
arXiv2014

On Kronecker Quotients

Leopardi introduced the notion of a Kronecker quotient in [Paul Leopardi. A generalized FFT for Clifford algebras. Bulletin of the Belgian Mathematical Society, 11:663--688, 2005.]. This article considers the basic properties that a Kronecker quotient should satisfy and additional properties which may be satisfied. A class of Kronecker quotients for which these properties have a natural description is completely characterized. Two examples of types of Kronecker quotients are described.

math.FA↗
arXiv2014

Exceptional Points, Nonnormal Matrices, Hierarchy of Spin Matrices and an Eigenvalue Problem

Exceptional points of a class of non-hermitian Hamilton operators $\hat H$ of the form $\hat H=\hat H_0+i\hat H_1$ are studied, where $\hat H_0$ and $\hat H_1$ are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators $\hat H_0$ and $\hat H_1$ are given by spin matrices for spin $s=1/2,1,3/2,\dots$. Since the linear operators studied are nonnormal, properties of such operators are described.

math-ph↗
arXiv2013

An Eigenvalue Problem for a Fermi System and Lie Algebras

We study a Fermi Hamilton operator $\hat K$ which does not commute with the number operator $\hat N$. The eigenvalue problem and the Schrödinger equation is solved. Entanglement is also discussed. Furthermore the Lie algebra generated by the two terms of the Hamilton operator is derived and the Lie algebra generated by the Hamilton operator and the number operator is also classified.

math-ph↗
arXiv2013

Cayley transform and the Kronecker product of Hermitian matrices

We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms.

math.FA↗