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 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 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).
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
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.
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}
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.
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.
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…
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.
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…
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…
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.
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…
