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# Some generalizations of commutativity for linear transformations on a finite dimensional vector space

## Citation

Gaines, Fergus John (1966) Some generalizations of commutativity for linear transformations on a finite dimensional vector space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PBNM-BZ75. https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548

## Abstract

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let Ai+1 = AiB - BAi, i = 0, 1, 2,…, with A = Ao. Let fk (A, B; σ) = A2K+1 - σ1A2K-1 + σ2A2K-3 -… +(-1)KσKA1 where σ = (σ1, σ2,…, σK), σi belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that fn(A, B; σ) = 0 if σi is the ith elementary symmetric function of (β4- βs)2, 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β4 are the characteristic roots of B. In this thesis we discuss relations involving fk(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that fk(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X1, X2…Xr belong to the radical of the algebra generated by A and B over F, where Xi has the form f2(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that fk(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of gk(w; σ) = w2K+1 - σ1w2K-1 + σ2w2K-3 - …. +(-1)K σKw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Item Type: Thesis (Dissertation (Ph.D.)) Mathematics California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Taussky-Todd, Olga Unknown, Unknown 4 April 1966 CaltechTHESIS:09222015-114033548 https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548 10.7907/PBNM-BZ75 No commercial reproduction, distribution, display or performance rights in this work are provided. 9165 CaltechTHESIS INVALID USER 25 Sep 2015 16:46 21 Dec 2019 03:05

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