Matrices with prescribed entries and characteristic polynomial

It is proved that there exists an n x n matrix over an arbitrary field with n 1 prescribed entries and prescribed characteristic polynomial. In [3] the author proves that there always exists an n x n matrix over an arbitrary field 0, with n 1 prescribed entries and prescribed eigenvalues in 0. Answering a question posed at the end of G. N. de Oliveira's paper, we shall solve the problem of finding a necessary and sufficient condition for the existence of an n x n matrix over 0, with n 1 prescribed entries and prescribed characteristic polynomial. Our principal results are the following: Theorem 1. Let a ... a1 E b and f(A) be a monic polynomial of degree n over b. Let (i1, ji) .*, (i n ins 1) be prescribed distinct positions in an n x n matrix. Suppose the following condition is not satis fied: (1) The prescribed positions are all the nonprincipal positions of a row or column and a1 = ... = an = 0. Then there exists an n x n matrix over b with at in the position (it, i t = 1,.**, n 1, and with /(AX) as characteristic polynomial. Theorem 2. With the same notation as in Theorem 1 assume that the condition (1) is satisfied. Then there exists an n x n matrix over b with at in the position (it, t) t = 1,..., n 1, and f(A) as characteristic polynomial if and only if f(A) has at least one root in b. Before bringing out the proofs of these theorems we will prove some auxiliary results. Let On be the vector space over 0 of the n x 1 matrices over 0. If v E dn and B is an n x n matrix over 0, the minimal polynomial of v relative Received by the editors November 17, 1972 and, in revised form, June 19, 1973. AMS (MOS) subject classifications (1970). Primary 15A15; Secondary 15A18. Copyright