Eigenvectors from eigenvalues revisited
英文摘要:
This is a remark on a recent post by P. Denton, S. Parke, T. Tao, X. Zhang, Eigenvectors from eigenvalues, arXiv:1908.03795v1
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Carlos Tomei
This is a remark on a recent post by P. Denton, S. Parke, T. Tao, X. Zhang, Eigenvectors from eigenvalues, arXiv:1908.03795v1
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
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.
The method of computing eigenvectors from eigenvalues of submatrices can be shown as equivalent to a method of computing the constraint which achieves specified stationary values of a quadratic optimization. Similarly, we show computation of eigenvectors of an orthonormal basis projection using eigenvalues of sub-projections.
I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit of Argand's proof of the fundamental theorem of algebra. The proof only relies on Weierstrass's theorem, the definition of the inverse of a linear operator and algebraic identities.
A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra
本文从外代数视角推广了特征向量-特征值恒等式,证明其成立当且仅当特征值的几何重数等于代数重数,并允许重数大于1,无需其他矩阵假设。
By using the methods of Cauchy-Binet type formula and adjugate matrix respectively, a wonderful equality relating to the elements of eigenvectors, the eigenvalues and the submatrix eigenvalues is proved in arXiv:1908.03795. In the note, we use matrix block to provide a new and shorter proof for the equality.
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…
A New Perspective on Eigenvalues and Eigenvectors of Bicomplex Linear Operators
该研究聚焦双复空间上双复线性算子的特征值与特征向量,探究相关算子性质、多组特征值特征向量存在条件定理,以及对应特征空间的结构与性质,为双复线性算子领域提供了新的研究视角。
This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then a real geometric interpretation is given to the eigenvalues and eigenvectors by means of real geometric algebra. The eigenvectors are seen to be \textit{two component eigenspinors} which can be further reduced to underlying vector duplets. The eigenvalues are interpreted as rotation operators, which rotate the underlying vector duplets. The second part of this paper extends and gener…
In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of eigenvalues of A and, thus, reduce the degree of the characteristic polynomial associated with this matrix. This reduction can occur in two ways: when restrictions between some elements of A are checked (or imposed) and/or when some eigenvalues of the submatrix of A are degenerate. As an application, we show how to obtain matrices of arbitrary size, not trivial,…
Generalized Eigenvectors and Rayleigh bounds for tropical algebraic eigenvalues
本文探讨了热带代数中特征对问题,提出广义热带特征向量概念,并通过计算方法构造这些向量,同时证明了热带矩阵特征值的上界。
Using the concept of (K,L)-eigenvector, we investigate the structure of the max-min eigenspace associated with a given eigenvalue of a matrix in the max-min algebra (also known as fuzzy algebra). In our approach, the max-min eigenspace is split into several regions according to the order relations between the eigenvalue and the components of x. The resulting theory of (K,L)-eigenvectors, being based on the fundamental results of Gondran and Minoux, allows to describe the whole max-min eigenspace explicitly and in more detail.
Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a convenient and powerful tool to succinctly determine eigenvectors from eigenvalues. The identity relates the eigenvector component to the eigenvalues of $A$ and the minor $M_j$, which is formulated in an elegant form as \[ \lvert v_{i,j} \rvert^2\prod_{k=1;k\ne i}^{n-1}({λ_i}(A)-{λ_k}(A))=\prod_{k=1}^{n-1}({λ_i}(A)-{λ_k}(M_j)). \,\,\,%\mbox{(\cite{tao-eig,D-…
On Universal Eigenvalues and Eigenvectors of Hypermatrices
本文提出超矩阵的通用特征向量概念,利用半张量积和首一分解算法将U-特征问题转化为一般矩阵特征问题,并揭示其几何意义与应用潜力。
For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.
We find the eigenvalues and eigenvectors of the n by n matrix with (i,j) entry \binom(i-1,n-j), establishing a conjecture of Peele and Stanica. Curiously, the eigenvectors can be chosen to form a matrix which is its own inverse.
If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $λ_1(A),\dots,λ_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $λ_i(A)$ is related to the eigenvalues $λ_1(M_j),\dots,λ_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(λ_i(A)-λ_k(A)\right)=\prod_{k=1}^{n-1}\left(λ_i(A)-λ_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be…
Block diagonal dominance of matrices revisited: bounds for the norms of inverses and eigenvalue inclusion sets
本文重新审视块对角占优矩阵的逆矩阵范数界,基于Feingold和Varga的条件推广,提出更紧的特征值包含区域。
A Note on the Eigenvalues of the Google Matrix
本文针对Google矩阵的特征值分布,给出了Langville-Meyer定理的另一种证明,该定理对幂法计算PageRank的收敛性分析具有重要意义。
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