Szilvia Homolya, Jeno Szigeti
We show, how the combination of the Cayley-Hamilton theorem and a certain companion matrix construction can be used to derive Z2-graded trace identities in M_{n}(E).
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We work to find a basis of graded identities for the octonion algebra. We do so for the $\mathbb{Z}_2^2$ and $\mathbb{Z}_2^3$ gradings, both of them derived of the Cayley-Dickson process, the later grading being possible only when the characteristic of the scalars is not two.
We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.
We develop the general Theory of Cayley Hamilton algebras and we compare this with the theory of pseudocharacters. We finally characterize prime $T$--ideals for Cayley Hamilton algebras and discuss some of their geometry. To the memory of T. A. Springer
We exhibit a Cayley-Hamilton trace identity for $2\times2$ matrices with entries in a ring $R$ satisfying $[[x,y],[x,z]]=0$ and 1/2 \in R$.
If A is an n \times n matrix over a ring R satisfying the polynomial identity [x,y][u,v]=0, then an invariant Cayley-Hamilton identity of the form ΣA^{i}c_{i,j}A^{j}=0 with c_{i,j}\in R and c_{n,n}=(n!)^2 holds for A.
For an nxn matrix A over a Lie nilpotent ring R of index k, we prove that an invariant "power" Cayley-Hamilton identity of degree (n^2)2^{k-2} holds. The right coefficients are not uniquely determined by A, and the cosets lambda_i+D, with D the double commutator ideal R[[R,R],R]R of R, appear in the so-called second right characteristic polynomial of the natural image of A in the nxn matrix ring M_{n}(R/D) over the factor ring R/D.
We introduce the notion of characteristic function of a quaternionic matrix, whose roots are the left eigenvalues. We prove that for all $2\times 2$ matrices and for $3\times 3$ matrices having some zero entry outside the diagonal there is a characteristic function which is a polynomial. For the other $3\times 3$ matrices the characteristic function is a rational function with one point of discontinuity. We prove that Hamilton-Cayley theorem holds in all cases.
Fermionic matrices and super Cayley--Hamilton algebras
本文通过发展经典情形的分次类比，建立了玻色子和费米子矩阵n元组的第一和第二基本定理。
On the left and right coefficients of the general Cayley-Hamilton identities for an nxn matrix
本文研究了n×n矩阵在任意单位环上的凯莱-哈密尔顿恒等式，证明了矩阵系数C(i)和D(i)的奇分量关系及奇分量的表达式。
We develop the general Theory of Cayley Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper $T$-ideals of Cayley Hamilton algebras, 2020, arXiv:2008.02222
We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, h…
We revisit the classical theorem by Cayley and Hamilton, "{\em each endomorphism is a root of its own characteristic polynomial}", from the point of view of {\em Hasse--Schmidt derivations on an exterior algebra}
We develop a theory of semialgebra Grassmann triples via Hasse-Schmidt derivations, which formally generalizes results such as the Cayley-Hamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.
Cayley Hamilton algebras and a paper by Skip Garibaldi
本文基于Skip Garibaldi的相关论文，针对域上有限维代数，开展了最小Cayley Hamilton范数的讨论研究。
Given an $n \times n$ nonsingular matrix A and the characteristic polynomial of A as the starting point, we will leverage the Cayley-Hamilton Theorem to efficiently calculate the maximal length Jordan Chains for each distinct eigenvalue of the matrix. Efficiency and speed are gained by seeking a certain type of starting vector as the first step of the algorithm. The method for finding this starting vector does not require calculating the $ker[(A-λI)^k]$ which is quite an expensive operation, and which is the usual approach taken in solving the Jordan Canonical basis problem. Given this startin…
This note concerns a one-line diagrammatic proof of the Cayley-Hamilton Theorem. We discuss the proof's implications regarding the "core truth" of the theorem, and provide a generalization. We review the notation of trace diagrams and exhibit explicit diagrammatic descriptions of the coefficients of the characteristic polynomial, which occur as the n+1 "simplest" trace diagrams. We close with a discussion of diagrammatic polarization related to the theorem.
We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pai…
Using the natural notion of {\em Hasse--Schmidt derivations on an exterior algebra}, we relate two classical and seemingly unrelated subjects. The first is the celebrated Cayley--Hamilton theorem of linear algebra, "{\em each endomorphism of a finite-dimensional vector space is a root of its own characteristic polynomial}", and the second concerns the expression of the bosonic vertex operators occurring in the representation theory of the (infinite-dimensional) Heinsenberg algebra.
The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTT-algebras as well as the Reflection equation algebras.
Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras
针对正交$O(k)$型量子矩阵代数，建立其凯莱-哈密顿定理类似形式，区分奇偶高度情况及偶数高度正、负分量版本，并用量子矩阵“特征值”参数化恒等式系数。
The Geometric Origin of the Cayley-Hamilton Theorem: A Constructive Proof via Dimensional Syzygy
本文通过维度syzygy的构造性证明，揭示Cayley-Hamilton定理的几何起源，并提供二维空间的验证及不依赖维度的证明方法。
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley-Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley-Hamilton algebra; its fully Azumaya locus is shown to contain a copy o…
The trace Cayley-Hamilton theorem
本文证明了矩阵迹、行列式及伴随矩阵的性质，包括迹凯莱-哈密尔顿定理，该定理表明对于任一n×n矩阵A，其特征多项式系数满足特定关系式。
The lowest discriminant ideal of a Cayley-Hamilton Hopf algebra
本文研究了Cayley-Hamilton Hopf代数的最低判别理想，用极大稳定模描述其零点集，并揭示其受winding自同构群作用支配，应用于群代数及量子代数等实例。
Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras
研究Cayley-Hamilton霍普夫代数的Chevalley性质与判别理想，证明判别理想零集的性质及霍普夫代数的Grothendieck环的Frobenius-Perron维度。
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