# Finite Dimensional Multilinear Algebra, Part II by Marvin Marcus

By Marvin Marcus

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**Example text**

As an ordered abelian group, we let M equal cok(I - A). We define the action of Lon M by defining the action of ton M to be the action of t n on the original module cok(I - A). 6 Small presentations. If a matrix is n x n, then we say it has size n. What is the smallest size of a matrix with a given nonzero spectrum? With a given shift equivalence class? These are difficult questions with unhappy answers. For example, consider the 4-tuple ()2, i, -i, E), where € is small and positive. 3]. 3]. For an example over the integers, consider the 3-tuple (5,1,1).

Springer-Verlag 1982. SACHS, Spectra of graphs, Academic Press (1980). E. COVEN & M. PAUL, Endomorphisms of irreducible subshifts of finite type, Math Systems Th. 8(1974), 167-175. CUNTZ, A class of C*-algebras and topological Markov chains II: reducible chains and the Ext-functor for C*-algebras, Inventiones Math. 63, 25-40 (1981). fur Reine und Angew. Math. 320 (1980), 44-51. KRIEGER, A class ofC*-Algebras and topological Markov chains, Inventiones Math. 56, 251-268 (1980). V. DE ANGELIS, Polynomial beta functions and positivity of polynomials, PhD.

Is the spectrum of a matrix A if the characteristic polynomial is XA(t) = Di(t - di ). ) Necessary conditions on ~ are discussed in [BH1], especially in Appendix 3. The best reference to the literature on this problem is still [BePI], for a more recent discussion see [Mi]. To my knowledge the problem first appears in print in Suleimanova's 1949 paper [Su] (if we neglect the glorious work of Perron and Frobenius early in this century). It has been rather intractable. The solution is known if n = 3 [LL] or if n = 4 and the entries of ~ are real [Ke].