Matrices with few nonzero principal minors

To generalize D-nilpotent matrices that play a role in study of Drukowski maps, we introduce quasi-D-nilpotent matrices. A matrix A is called quasi-D-nilpotent if there exists a subspace V of diagonal matrices of codimension 1 such that DA is nilpotent for all . It is proved that a quasi-D-nilpotent matrix has few nonzero principal minors. We also determine irreducible quasi-D-nilpotent matrices and the Frobenius normal forms of quasi-D-nilpotent matrices with respect to permutation similarity.