# Combinatorial and Graph-Theoretical Problems in Linear by Mike Boyle (auth.), Richard A. Brualdi, Shmuel Friedland,

By Mike Boyle (auth.), Richard A. Brualdi, Shmuel Friedland, Victor Klee (eds.)

This IMA quantity in arithmetic and its functions COMBINATORIAL AND GRAPH-THEORETICAL difficulties IN LINEAR ALGEBRA relies at the complaints of a workshop that was once an essential component of the 1991-92 IMA application on "Applied Linear Algebra." we're thankful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for making plans and imposing the year-long application. We specially thank Richard Brualdi, Shmuel Friedland, and Victor Klee for organizing this workshop and enhancing the lawsuits. The monetary aid of the nationwide technology beginning made the workshop attainable. A vner Friedman Willard Miller, Jr. PREFACE The 1991-1992 application of the Institute for arithmetic and its functions (IMA) was once utilized Linear Algebra. As a part of this software, a workshop on Com binatorial and Graph-theoretical difficulties in Linear Algebra was once hung on November 11-15, 1991. the aim of the workshop was once to assemble in an off-the-cuff environment the varied crew of people that paintings on difficulties in linear algebra and matrix thought during which combinatorial or graph~theoretic research is a big com ponent. a few of the individuals of the workshop loved the hospitality of the IMA for the complete fall region, during which the emphasis was once discrete matrix analysis.

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**Combinatorial and Graph-Theoretical Problems in Linear Algebra**

This IMA quantity in arithmetic and its functions COMBINATORIAL AND GRAPH-THEORETICAL difficulties IN LINEAR ALGEBRA is predicated at the lawsuits of a workshop that was once a vital part of the 1991-92 IMA application on "Applied Linear Algebra. " we're thankful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for making plans and imposing the year-long software.

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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].