Matrices with prescribed eigenvalues and prescribed submatrices II

摘要

Let F be a field and let n,p1,p2,p3 be positive integers such that n=p1+p2+p3. LetC=C1,1C1,2C1,3C2,1C2,2C2,3C3,1C3,2C3,3∈Fn×n,where the blocks Ci,j∈Fpi×pj,i,j∈{1,2,3}. In this paper we describe the eigenvalues of C, when C1,2,C1,3, C2,2 are prescribed (with C1,3=0) and the other blocks are unknown. For the same prescription of blocks (for arbitrary prescription of C1,3), we still provide a sufficient condition for the prescription of the characteristic polynomial of C.