Tevian Dray, Jason Janesky, Corinne A. Manogue
We extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigenvalues are not real, giving a complete treatment of the 2x2 case, and summarizing some prelimenary results for the 3x3 case.
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We discuss our preliminary attempts to extend previous work on 2x2 Hermitian octonionic matrices with non-real eigenvalues to the 3x3 case.
We discuss the eigenvalue problem for 2x2 and 3x3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.
We show that any 3-component octonionic vector which is purely imaginary, but not quaternionic, is an eigenvector of a 6-parameter family of Hermitian octonionic matrices, with imaginary eigenvalue equal to the associator of its elements.
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the u…
We revisit the octonionic eigenvalue problem from a geometric perspective. In particular, we study a tautological sheaf defined on a sextic related to this problem, the Ogievetskiî-Dray-Manogue sextic. We then define and study a twisted version of the octonionic eigenvalue problem. A new sextic arises in this setting and we study the corresponding tautological sheaf supported on it. This twisted version of the octonionic eigenvalue problem is eminently more symmetric than the original one, as reflected by the last result we prove in this paper : the automorphism group of the twisted octonionic…
The Hermitian decomposition of a linear operator is generalized to the case of two or more operations. An additive expansion of the product of three octonions into three parts is constructed, wherein each part either preserve or change the sign under the action of the Hermitian conjugation and operation of inversion of the multiplicative order of three hypercomplex numbers, as well as under the composition of specified operations. The product of three octonions, in particular quaternions, with conjugate central factor is presented as the sum of mutually orthogonal triple anticommutator, triple…
We study analogues of classical inequalities for the eigenvalues of sums of pseudo-Hermitian matrices.
Denton, Parke, Tao and Zhang gave a new method which determines eigenvectors from eigenvalues for Hermitian matrices with distinct eigenvalues. In this short note, we extend the above result to general Hermitian matrices.
We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive semi-definite or definite if and only if all of its right eigenvalues and subeigenvalues are nonnegative or positive, respectively. A Hermitian matrix can be diagonalized if and only if it has no right subeigenvalues. Then we present the singular value decomposition for general dual complex matrices. The results are further extended to dual quaternion matrices.
We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices.
We discuss the eigenvalue problem for 3x3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and solve the usual characteristic equation. We give an elementary construction of the corresponding eigenmatrices, and we further speculate on a possible application to particle physics.
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserve…
We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.
The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual quaternion matrices. If a right eigenvalue is a dual number, then it is also a left eigenvalue. In this case, this dual number is called an eigenvalue of that dual quaternion matrix. We show that the right eigenvalues of a dual quaternion Hermitian matrix are dual numbers. Thus, they are eigenvalues. An $n \times n$ dual quaternion Hermitian matrix is shown to h…
We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of characteristic polynomial that collects all eigenvalues that have the same multiplicity into one polynomial factor with coefficients rationally expressed through coefficients of original characteristic polynomial. Finding with any accuracy a minimal distance between eigenvalues, as well as a minimal and a maximal eigenvalues of the Hermitian operator, applying only ra…
Eigenvalues of Dual Hermitian Matrices with Application in Formation Control
本文提出补充矩阵方法计算对偶Hermitian矩阵特征值，并将其与单位增益图理论结合，为多智能体编队控制中的相对构型可行性问题提供跨学科求解途径。
The Nearest Hermitian Inverse Eigenvalue Problem Solution with Respect to the 2-Norm
本文研究在给定埃尔米特矩阵的特征值和给定埃尔米特矩阵的情况下，如何找到最接近给定算子的埃尔米特矩阵，通过随机矩阵和灰度图像测试其平滑特性。
Octonionic structure operator and its right spectrum
研究仅用八元数乘法定义的规范$G_2$-等变算子$h$的实谱和八元数右特征值问题，通过$G_2$分解和$\mathrm{SU}(3)$块分解得到谱轨迹。
Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C <= A + B are given by the same inequalities as in Klyachko's theorem for the case where C = A + B, except that the equality corresponding to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C) <= tr(A) + tr(B). The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B -> C -> A are characterized by the same inequalities.
Aggregate Bounds on the eigenvalues of the principal submatrices of a Hermitian matrix and majorization relations
本文扩展了Thompson关于Hermitian矩阵主子矩阵特征值和的界，并利用该结果推导出所有主子矩阵特征值之间的主导关系，涵盖Schur和Szasz的已有结果。
Given a diagonalizable $N\times N$ matrix $H$, whose non-degenerate spectrum consists of $p$ pairs of complex conjugate eigenvalues and additional $N-2p$ real eigenvalues, we determine all metrics $M$, of all possible signatures, with respect to which $H$ is pseudo-hermitian. In particular, we show that any compatible $M$ must have $p$ pairs of opposite eigenvalues in its spectrum so that $p$ is the minimal number of both positive and negative eigenvalues of $M$. We provide explicit parametrization of the space of all admissible metrics and show that it is topologically a $p$-dimensional torus…
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