**6. Some open questions**

In [3, Theorem 1], Minc proved the following result:

**Theorem 16** [3]*Let Λ* = {*λ*1, *λ*2, …, *λn*} *be a list of complex numbers, which is realizable by a diago‐ nalizable positive matrix A*. *Then, for each JCF J<sup>Λ</sup> associated with Λ*, *there exists a positive matrix B with the same spectrum as A*, *and with JCF J*(*B*) = *JΛ*.

According to Minc, the positivity condition is essential in his proof, and it is not known if the result holds without this condition (see [2]). Specifically, it is not known *i*) whether for every positive matrix, there exists a diagonalizable positive matrix with the same spectrum, *ii*) whether for every nonnegative diagonalizable matrix with spectrum *Λ* = {*λ*1, …, *λn*}, there exists a nonnegative matrix for each *JCF* associated with *Λ*. There are many examples which show that the Minc result holds for diagonalizable nonnegative matrices. For instance, all diago‐ nalizable nonnegative matrices with spectrum *Λ* given in Section 2, Theorems 4 and 5, and Corollaries 1 and 2, give rise to nonnegative matrices for each one of the possible *JCF* associated with *Λ*. However, we do not know if the Minc result holds for a general diagonalizable nonnegative matrix.

The problem of finding a nonnegative matrix with prescribed spectrum and diagonal entries is a hard open problem, which, besides of being important in itself, is necessary to apply dif‐ ferent versions of Rado's Theorem on both problems, the *NIEP* and the *NIEDP*. Necessary and sufficient conditions are only known for *n* ≤ 3.
