R. Merris, Two problems involving Schur functions, Linear Algebra Appl. 10 (1975), 155-162. AMS Notices 22 (1975), A-426. Math. Reviews 51 # 542. The author is grateful for the generous hospitality of the Instituto de Fisica e Matematica during the preparation of this article.

A scalar valued function d of the m-by-m matrices is defined, depending on a unitary representation of degree k of a permutation group G of degree m. The questions are these: (1) What can be said about the roots of d(zI - A)? and (2) Suppose A, B, and A - B are positive semidefinite hermitian matrices. If d(A) = d(B) >< 0, must A = B? Some partial answers are obtained.

Similar work can be found in

  • R. Merris and J. A. Dias da Silva, J. Algebra 35 (1975), 442-448.
  • R. Merris, Linear Algebra Appl. 20 (1978), 77-86.
  • M. Ishaq and R. Merris, Linear Algebra Appl. 25 (1979), 255-269.

    • G. H. Chan and M. H. Lim [A conjecture involving generalized matrix functions, Linear Algebra Appl. 36 (1981), 157-163] obtained an affirmative answer to Question 2 when n = k.

    Among the publications citing this article are:

  • M. Marcus, Finite Dimensional Multilinear Algebra Part II, Marcel Dekker, NY, 1975.
  • D. K. Baxter, Linear Algebra Appl. 22 (1978), 175-189.
  • H. Minc, Permanents, v. 6, Encyclopedia of Mathematics and Appl., Addison-Wesley, Reading, MA, 1978.
  • G. N. de Oliveira, Linear & Multilinear Algebra 12 (1982), 125-138.
  • G. N. de Oliveira and J. A. Dias da Silva, Linear Algebra Appl. 49 (1983), 191-219.
  • A. Krauter, Linear Algebra Appl. 93 (1987), 39-55.
  • V. Strok, Permanents: Theory and Appl. [Russian], Krasnoyarsk Politekhn. Inst., 1990, 73-78.