**Author details**

Ricardo L. Soto

( )

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0 10 90 0 <sup>45</sup> 3 1 0 31

ê ú é ù ê ú ê ú <sup>=</sup> = - ê ú ê ú ê ú ê ú

00 3 <sup>45</sup> 81 0

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<sup>0030</sup> <sup>2</sup> <sup>45</sup> 1 1 <sup>=</sup> 01 0 0 0300 1 <sup>2</sup> 2 2 0 0 <sup>0000</sup> <sup>2</sup> <sup>45</sup> 00 01 81 0 001 2

90 0 0 <sup>45</sup> <sup>1</sup> 20 3 2

**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*

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

*J A*


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2

*Then,*

112 Applied Linear Algebra in Action

*A*

**6. Some open questions**

nonnegative matrix.

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*B with J B*

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4 2 45 45 81 2 2 0 4 4

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

*with the same spectrum as A*, *and with JCF J*(*B*) = *JΛ*.

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Address all correspondence to: rsoto@ucn.cl

Department of Mathematics, Universidad Católica del Norte, Antofagasta, Chile
