Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

We show that for every invertible n×n complex matrix A there is an n×n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n×n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n-1 are zero.