**2. Cases completely solved**

In this section, we shall discuss about certain subproblems of the *NIEDP*, for which a complete solution has been obtained. We start with a list of real nonnegative numbers *Λ* = {*λ*1, *λ*2, …, *λn*} with *λ*1 > *λ*2 ≥ ⋯ ≥ *λn* ≥ 0. Let *D* = *diag*{*λ*1, *λ*2, …, *λn*}. In [17], Perfect shows that the matrix


has the property that *PDP* <sup>−</sup>1∈CS*λ*<sup>1</sup> is positive diagonalizable with spectrum *Λ*. Then we have:

**Theorem 4** [6] *Let Λ* = {*λ*1, …, *λn*} *be a list of real numbers with λ*1 > *λ*<sup>2</sup> ≥ ⋯ ≥ *λ<sup>n</sup>* ≥ 0. *Then, there exists a nonnegative matrix A*∈CS*λ*<sup>1</sup> *with spectrum Λ*, *for each possible JCF associated with Λ*.

**Proof**. Let *D* = *diag*{*λ*1, *λ*2, …, *λn*} and let *P* be the matrix in (2). Then *PDP* <sup>−</sup>1∈CS*λ*<sup>1</sup> is positive (generalized stochastic) with spectrum *Λ* and linear elementary divisors. Let

$$K = \{2, 3, \ldots, n - 1\} \text{ and } E = \sum\_{i \in K} E\_{i, i+1},$$

[ ] <sup>1</sup> 1 2 1 1 1 2

*S XCS C C S*

and we have *σ*(*A* + *XC*) = *σ*(*Ω* + *CX*) + *σ*(*A*) − *σ*(*Ω*).■

*i*=1 *n qi* ≠*λ<sup>i</sup>*

*qi* =0, *then A and A* + **eq***<sup>T</sup> are similar.*

**2. Cases completely solved**

has the property that *PDP* <sup>−</sup>1∈CS*λ*<sup>1</sup>

Brauer perturbation *A* + **eq***<sup>T</sup>*.

*Let***q***<sup>T</sup>* = (*q*1, …, *qn*) *and <sup>λ</sup>*<sup>1</sup> <sup>+</sup> ∑

**Lemma 1** [6] *Let A*∈CS*λ*<sup>1</sup> *be with JCF*

*S A XC S S AS S XCS*

Thus,

90 Applied Linear Algebra in Action

*if* ∑ *i*=1 *n*

2 2

*CX UAY CY*

ë û

*VAY*

 l

is positive diagonalizable with spectrum *Λ*. Then we have:

*i*=1 *n*

*qi*)*E*11.*In particular,*

(2)

is positive

*X Y*



( ) <sup>1</sup> 1 1 , <sup>0</sup>


The following result in [6], which will be frequently used later, shows how is the *JCF* of the

( ) { () ( ) ( ) <sup>2</sup> } <sup>1</sup> 11 2 , ,, *<sup>k</sup> <sup>n</sup> n k J A S AS diag J J J*

In this section, we shall discuss about certain subproblems of the *NIEDP*, for which a complete solution has been obtained. We start with a list of real nonnegative numbers *Λ* = {*λ*1, *λ*2, …, *λn*} with *λ*1 > *λ*2 ≥ ⋯ ≥ *λn* ≥ 0. Let *D* = *diag*{*λ*1, *λ*2, …, *λn*}. In [17], Perfect shows that the matrix

> 11 1 1 11 1 1

L L

é ù ê ú -

1 10 0

ë û -

**Theorem 4** [6] *Let Λ* = {*λ*1, …, *λn*} *be a list of real numbers with λ*1 > *λ*<sup>2</sup> ≥ ⋯ ≥ *λ<sup>n</sup>* ≥ 0. *Then, there exists*

1 1

= -

*a nonnegative matrix A*∈CS*λ*<sup>1</sup> *with spectrum Λ*, *for each possible JCF associated with Λ*.

**Proof**. Let *D* = *diag*{*λ*1, *λ*2, …, *λn*} and let *P* be the matrix in (2). Then *PDP* <sup>−</sup>1∈CS*λ*<sup>1</sup>

(generalized stochastic) with spectrum *Λ* and linear elementary divisors. Let

MML

*P*

1 0

OO M L


ll

, *i* = 2, …, *n*. *Then the JCF of A* + **eq***<sup>T</sup> is J*(*A*) + (∑

such that *D* + *E* is the desired *JCF*. Then there exists *ε* > 0 such that *A* = *PDP*<sup>−</sup> 1 + *εPEP*<sup>−</sup> 1 is nonnegative, and since *D* + *εE* and *D* + *E* are diagonally similar (with *diag*{1, *ε*, *ε*<sup>2</sup> , …, *ε<sup>n</sup>* − 1}), *A* has *JCF* equal to *D* + *E*. Moreover, since *P* **e**1 = **e**, the rows of *P*<sup>−</sup><sup>1</sup> satisfy (1) and *PEP*<sup>−</sup><sup>1</sup> **e** = 0. Thus *A*∈CS*λ*<sup>1</sup> has the desired elementary divisors. ■

The following two cases correspond, respectively, to lists *Λ* = {*λ*1, …, *λn*} of real numbers sat‐ isfying *λ*1 > 0 > *λ*<sup>2</sup> ≥ ⋯ ≥ *λn*, which are called lists of Suleimanova type; and to lists *Λ* = {*λ*1, …, *λn*} of complex numbers satisfying *λ<sup>i</sup>* ∈ ℱ, *i* = 2, …, *n*, where

$$F = \left\{ z \in C : \text{Re } z < 0, \left| \text{Re } z \right| \ge \left| \text{Im } z \right| \right\},$$

which are called lists of complex Suleimanova type. In both cases, the *NIEP* has a solution if and only if ∑ *i*=1 *n λ<sup>i</sup>* ≥0 (see [18, 19]).

Let *Λ* = {*λ*1, …, *λn*} be a list of complex numbers with *Λ*¯ <sup>=</sup>*Λ*, <sup>∑</sup> *i*=1 *n λ<sup>i</sup>* ≥0, and *λ*1 ≥ |*λ<sup>i</sup>* |, *i* = 2, …, *n*. In [9, Theorem 2.1], the authors define

$$\begin{aligned} M &= \max\_{2 \le k \le n} \{ 0, Re\mathcal{A}\_{\boldsymbol{\lambda}} + Im\mathcal{A}\_{\boldsymbol{\lambda}} \} \\ m &= -\sum\_{k=2}^{n} \min \left\{ 0, Re\mathcal{A}\_{\boldsymbol{\lambda}}, Im\mathcal{A}\_{\boldsymbol{\lambda}} \right\} \end{aligned}$$

and show that if *λ*1 ≥ *M* + *m*, when all possible Jordan blocks *Jni* (*λi* ), of size *ni* ≥ 2, are associated to a real eigenvalue *λ<sup>i</sup>* < 0; or when there is at least one Jordan block *Jni* (*λi* ), of size *ni* ≥ 2, associated to a real eigenvalue *λ<sup>i</sup>* ≥ 0 with *M* =R*eλ<sup>i</sup>* 0 + I*mλ<sup>i</sup>* 0 , for some *i*0, then there exists an *n* × *n* nonnegative matrix *A*∈CS*λ*<sup>1</sup> with spectrum *Λ* and with prescribed elementary divisors. The result in [9, Theorem 2.1] was used by the authors to prove the following

**Corollary 1** [9] *Let Λ* = {*λ*1, …, *λn*} *be a list of complex numbers with λ<sup>i</sup>* ∈ ℱ, *i* = 2, …, *n*. *Then, for each possible JCF associated with Λ*, *there exists a nonnegative matrix A*∈CS*λ*<sup>1</sup> *with spectrum Λ if and only if* ∑ *n λ<sup>i</sup>* ≥0.

**Proof**. The condition is necessary for the existence of a nonnegative matrix with spectrum *Λ*. The condition is also sufficient. In fact, since

*i*=1

$$M = \max\_{2 \le \lambda \le n} \left\{ 0, \operatorname{Re} \vec{\lambda}\_{\boldsymbol{\lambda}} + \operatorname{Im} \vec{\lambda}\_{\boldsymbol{\lambda}} \right\} = 0,$$

$$\dim = -\sum\_{k=2}^{n} \min \left\{ 0, \operatorname{Re} \vec{\lambda}\_{\boldsymbol{\lambda}}, \operatorname{Im} \vec{\lambda}\_{\boldsymbol{\lambda}} \right\} = -\sum\_{k=2}^{n} \operatorname{Re} \vec{\lambda}\_{\boldsymbol{\lambda}} = -\sum\_{k=2}^{n} \vec{\lambda}\_{\boldsymbol{\lambda}}.$$

then if *λ*1 + *λ*2 + ⋯ + *λn* ≥ 0, *λ*<sup>1</sup> ≥ −∑ *k*=2 *n λ<sup>i</sup>* =*M* + *m*. ■

In [19, Theorem 3.3], the authors show that if *Λ* = {*λ*1, …, *λn*} is a list of complex numbers with *λ<sup>i</sup>* ∈ ℱ, *i* = 2, …, *n*, then there exists a nonnegative matrix with spectrum *Λ* if and only if ∑ *i*=1 *n λ<sup>i</sup>* ≥0. Thus we have:

**Corollary 2** [6] *Let Λ* = {*λ*1, …, *λn*} *be a list of real numbers with λ*1 > 0 > *λ*2 ≥ ⋯ ≥ *λn*. *Then, for each possible JCF associated with Λ*, *there exists a nonnegative matrix A*∈CS*λ*<sup>1</sup> *with spectrum Λ if and only*

$$\left\| f \sum\_{i=1}^{n} \lambda\_i \right\| \ge 0.$$

In [20], Šmigoc extends the result in [19, Theorem 3.3], to the region

$$\mathcal{G} = \{ \mathbf{z} \in \mathbb{C} \colon \mathbf{Re} \mid \mathbf{z} < 0, |\sqrt{3}\mathbf{Re} \mid \mathbf{z}| \ge |\text{Im } \mathbf{z}| \},$$

that is, she shows that if *Λ* = {*λ*1, …, *λn*} is a list of complex numbers with *λ<sup>i</sup>* ∈G, *i* = 2, …, *n*, then there exists a nonnegative matrix with spectrum *Λ* if and only if ∑ *i*=1 *n λ<sup>i</sup>* ≥0. The following result extends Corollary 1 to lists of complex numbers *Λ* = {*λ*1, …, *λn*}, with *λ<sup>i</sup>* ∈G, *i* = 2, …, *n*. Before, to state this result and its proof, we need the following lemmas, which we set here without proof:

**Lemma 2** [20] *Let A*= *A*<sup>1</sup> **a b***<sup>T</sup> c be an n* × *n matrix and let B be an m* × *m matrix with JCF <sup>J</sup>*(*B*)= *<sup>c</sup>* **<sup>0</sup> <sup>0</sup>** *<sup>I</sup>*(*B*) .

*Then the matrix*

$$C = \begin{bmatrix} A\_1 & \mathbf{a}\mathbf{t}^T \\ \mathbf{s}\mathbf{b}^T & B \end{bmatrix}, B\mathbf{s} = \mathbf{c}\mathbf{s}, \mathbf{t}^T B = \mathbf{c}\mathbf{t}^T, \text{with} \mathbf{t}^T \mathbf{s} = \mathbf{1}, \mathbf{0}$$

*has JCF*

$$J(C) = \begin{bmatrix} J(A) & 0 \\ 0 & I(B) \end{bmatrix}.$$

**Lemma 3** [12] *Let Λ* = {*λ*1, *a* ± *bi*, …, *a* ± *bi*} *be a list of n complex numbers, with n* ≥ 3, *a* < 0, *b* > 0. *If*

$$\lambda\_1 \ge \max\left\{-\left(n-1\right)a, -\frac{\left(2n-5\right)a^2 + b^2}{2a}\right\},\tag{3}$$

*then for each JCF associated with Λ*, *there exists an n* × *n nonnegative matrix A* = (*aij*), *with spectrum Λ and entry ann* = *s*1(*Λ*).

**Corollary 3** [12] *Let Λ* = {*λ*1, *a* ± *bi*, …, *a* ± *bi*} *be a list of n complex numbers, with* 0 < *b* ≤ − 3*a*. *Then, for each JCF associated with Λ*, *there exists a nonnegative matrix A* = (*aij*) *with spectrum Λ and entry ann* = *s*1(*Λ*), *if and only if λ*1 + (*n* − 1)*a* ≥ 0.

**Proof**. It is clear that the condition is necessary. Assume that *λ*1 + (*n* − 1)*a* ≥ 0. Since 0 < *b* ≤ − 3*a*, it follows that

$$\begin{aligned} \max \left\{ -\left(n-1\right)a, -\frac{\left(2n-5\right)a^2 + b^2}{2a} \right\} \\\\ \text{s.t.} \\\\ = \max \left\{ -\left(n-1\right)a, -\left(n-1\right)a + \frac{3a^2 - b^2}{2a} \right\} \\\\ = -\left(n-1\right)a \\\\ \leq \lambda\_1. \end{aligned}$$

Hence, from Lemma 3, *Λ* is the spectrum of a nonnegative matrix *A* = (*aij*) with entry *ann* = *s*1(*Λ*), for each *JCF* associated with *Λ*.

Now we state the main result of this section:

**Theorem 5** [12] *Let Λ* = {*λ*1, …, *λn*} *be a list of complex numbers with λ<sup>i</sup>* ∈G, *i* = 2, …, *n*. *Then, for each JCF associated with Λ*, *there exists a nonnegative matrix A with spectrum Λ and entry ann* <sup>=</sup>∑ *i*=1 *n λi*

$$\text{if } \text{and } \text{only } \text{if } \sum\_{i=1}^{n} \lambda\_i \ge 0.$$

{ } <sup>2</sup> max 0, 0, *k k k n*

2 2 2 min 0, , R , *n n n*

= = = = -å åå =- =-

*k k k*

In [19, Theorem 3.3], the authors show that if *Λ* = {*λ*1, …, *λn*} is a list of complex numbers with

**Corollary 2** [6] *Let Λ* = {*λ*1, …, *λn*} *be a list of real numbers with λ*1 > 0 > *λ*2 ≥ ⋯ ≥ *λn*. *Then, for each possible JCF associated with Λ*, *there exists a nonnegative matrix A*∈CS*λ*<sup>1</sup> *with spectrum Λ if and only*

that is, she shows that if *Λ* = {*λ*1, …, *λn*} is a list of complex numbers with *λ<sup>i</sup>* ∈G, *i* = 2, …, *n*, then

extends Corollary 1 to lists of complex numbers *Λ* = {*λ*1, …, *λn*}, with *λ<sup>i</sup>* ∈G, *i* = 2, …, *n*. Before, to state this result and its proof, we need the following lemmas, which we set here without

<sup>1</sup> , , ,with 1,

é ù = == = ê ú

*λ<sup>i</sup>* ∈ ℱ, *i* = 2, …, *n*, then there exists a nonnegative matrix with spectrum *Λ* if and only if ∑

= +=

 l

*kk k i*

 l  l

*i*=1 *n*

*be an n* × *n matrix and let B be an m* × *m matrix with JCF <sup>J</sup>*(*B*)= *<sup>c</sup>* **<sup>0</sup>**

*TT T*

**s st t ts**

( ) <sup>0</sup> . <sup>0</sup>

*I B* é ù <sup>=</sup> ê ú ë û *λ<sup>i</sup>* ≥0. The following result

**<sup>0</sup>** *<sup>I</sup>*(*B*) .

*i*=1 *n λ<sup>i</sup>* ≥0.

*M Re Im* l

{ }

*m Re Im e* ll

*λ<sup>i</sup>* =*M* + *m*. ■

£ £

*k*=2 *n*

In [20], Šmigoc extends the result in [19, Theorem 3.3], to the region

there exists a nonnegative matrix with spectrum *Λ* if and only if ∑

*T*

*J C*

*<sup>A</sup> <sup>C</sup> B c Bc B*

( ) ( )

*J A*

**Lemma 3** [12] *Let Λ* = {*λ*1, *a* ± *bi*, …, *a* ± *bi*} *be a list of n complex numbers, with n* ≥ 3, *a* < 0, *b* > 0. *If*

*T*

**sb**

ë û **at**

*A*<sup>1</sup> **a b***<sup>T</sup> c*

then if *λ*1 + *λ*2 + ⋯ + *λn* ≥ 0, *λ*<sup>1</sup> ≥ −∑

92 Applied Linear Algebra in Action

Thus we have:

*if* ∑ *i*=1 *n λ<sup>i</sup>* ≥0.

proof:

*has JCF*

**Lemma 2** [20] *Let A*=

*Then the matrix*

**Proof**. It is clear that the condition is necessary. Let *λ*1 > 0 > *λ*2 ≥ ⋯ ≥ *λ<sup>p</sup>* be real numbers and let *<sup>λ</sup>p*+1, *<sup>λ</sup>p*+2 <sup>=</sup>*λ*¯ *<sup>p</sup>*+1, …, *<sup>λ</sup>n*−1, *<sup>λ</sup><sup>n</sup>* <sup>=</sup>*λ*¯ *<sup>n</sup>*−1 be complex nonreal numbers, with *s*<sup>1</sup> (*Λ*)=∑ *i*=1 *n λ<sup>i</sup>* ≥0. Let *m* be the number of distinct pairs of complex conjugate. Consider the partition

$$
\Lambda\_{\mathbf{o}} = \left\{ \mathcal{A}\_1, \mathcal{A}\_2, \dots, \mathcal{A}\_p \right\}
$$

$$
\Lambda\_r = \left\{ \mathcal{S}\_1(\Lambda\_{r-1}), \mathcal{A}\_{p+r}, \overline{\mathcal{A}}\_{p+r}, \dots, \mathcal{A}\_{p+r}, \overline{\mathcal{A}}\_{p+r} \right\}, r = 1, \dots, m
$$

with *Λr* having *nr* + 1 elements, where *nr* is the number of elements of *Λr* − {*s*1(*Λ<sup>r</sup>* − 1)}, in such a way that ∑ *r*=1 *m nr* =*n* − *p*. The list *Λ*0 satisfies Corollary 2, and then, we can compute a matrix *A*<sup>0</sup> =(*aij* (0) ) with spectrum *Λ*0, the entry *app* (0) =*s*1(*Λ*0) and with arbitrarily prescribed elementary divisors. Moreover, since 0<Im *λp*+*<sup>r</sup>* ≤ − 3Re *λp*+*r*, *r* = 1, …, *m*, there exists, from Corollary 3, a nonneg‐ ative matrix *Ar* =(*aij* (*r*) ) of size *nr* + 1, with spectrum *Λr*, entry *anr*+1,*nr*+1 (*r*) <sup>=</sup>*s*1(*Λr*), for each *JCF* associated with *Λr*. Finally, by employing the Lemma 2 (Šmigoc Lemma) *m* times, we construct a nonnegative matrix *A* with spectrum *Λ* and with arbitrarily prescribed elementary divisors. ■

**Example 1***Let*

$$\Lambda = \{23, -1, -1, -1, -2 \pm 3i, -2 \pm 3i, -3 \pm 5i, -3 \pm 5i\}$$

*be given. We want to construct a* 12 × 12 *nonnegative matrix with elementary divisors*

( )( ) ( ) ( ) <sup>2</sup> <sup>2</sup> <sup>2</sup> l l l ll - + + ++ 23 , 1 , 1 , 4 13 ( )( )( )( ) llll+- +- ++ ++ 3 5, 3 5, 3 5, 3 5. *iiii*

*Consider the lists*

$$
\Lambda\_0 = \{23, -1, -1, -1\}
$$

$$
\Lambda\_1 = \{20, -2 \pm 3i, -2 \pm 3i\}
$$

$$
\Lambda\_2 = \{12, -3 \pm 5i, -3 \pm 5i\}.
$$

*Clearly Λ*0*is realizable by a nonnegative matrix A*0, *with elementary divisors* (*λ* − 23), (*λ* + 1)2 , (*λ* + 1), *and entry a*<sup>44</sup> (0) =20. *Λ*<sup>1</sup> *and Λ*<sup>2</sup> *are also realizable by nonnegative matrices A*1, *with elementary divisors*

(*λ* − 20), (*λ*<sup>2</sup> + 4*λ* + 13)2 , *and entry a*<sup>55</sup> (1) =12, *and A*2, *with linear elementary divisors* (*λ* − 12), (*λ* + 3 ± 5*i*), (*λ* + 3 ± 5*i*), *and entry a*<sup>55</sup> (2) =0, *respectively (see Corollary 3). By applying Lemma 2 to the matrices A*<sup>0</sup> *and A*<sup>1</sup> *we obtain an* 8 × 8 *nonnegative matrix B*1 = (*bij*) *with spectrum Λ*<sup>0</sup> ∪ *Λ*<sup>1</sup> − {20}, *elementary divisors*

$$\left(\begin{pmatrix}\mathcal{A}-\mathfrak{D}\mathfrak{Z}\end{pmatrix},\begin{pmatrix}\mathcal{A}+\mathfrak{I}\end{pmatrix}^{\circ},\begin{pmatrix}\mathcal{A}+\mathfrak{I}\end{pmatrix},\begin{pmatrix}\mathcal{A}^{\circ}+\mathfrak{A}\mathcal{A}+\mathfrak{I}\mathfrak{Z}\end{pmatrix}^{\circ}\right)$$

*and entry b*88 = 12. *Next, we employ Lemma 2 again with the matrices B*<sup>1</sup> *and A*2, *to obtain a nonnegative matrix A with spectrum Λ and the prescribed elementary divisors.*

The method that we employ to prove Theorem 5, allow us to compute, under certain condi‐ tions, a nonnegative matrix *A* with spectrum *Λ* = {*λ*1, …, *λn*}, where *λ<sup>i</sup>* ∈G, *i* = 2, …, *n*, for each *JCF* associated with *Λ*. However, this procedure does not work for any list of complex numbers in the left half plane. In particular, the list

$$\Lambda = \left\{6, -1+3i, -1-3i, -1+3i, -1-3i\right\}$$

satisfies the Laffey and Šmigoc conditions [21], and therefore, it is realizable. However, *Λ* does not satisfy the condition of Lemma 3. As a consequence, we cannot obtain, from Lemma 3, a nonnegative matrix with spectrum *Λ* and linear elementary divisors.
