**5. NIEDP for persymmetric matrices**

In this section, we consider the *NIEDP* for persymmetric matrices. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations, where the eigenvalues of the Gram matrix, which is symmetric and persymmetric, play an important role [22]. As the superscript *T*, in *AT*, denotes the transpose of *A*, the superscript *F*, in *AF*, denotes the flip-transpose of *A*, which flips *A* across its skew-diagonal. If *A* = (*aij*)*mn*, then *AF* = (*an* − *j* + 1,*m* − *<sup>i</sup>* + 1)*nm*. A matrix *A* is said to be *persymmetric* if *AF* = *A*, that is, if it is symmetric across its lower-left to upper-right diagonal. Let *J* be the *n* × *n* matrix with ones along its skew-diagonal and zeroes elsewhere, that is, *J* = [**e***n*|**e***<sup>n</sup>* <sup>−</sup> 1| ⋯ |**e**1]. Then

$$J^{\mathcal{T}} = J^{\mathcal{F}} = J, \; J^2 = I, \; A^{\mathcal{F}} = J A^{\mathcal{T}} J,$$

and the following properties are straightforward:

$$\left(\left(A^{F}\right)^{F} = A, \left(A^{T}\right)^{F} = \left(A^{F}\right)^{T}, \left(A+B\right)^{F} = A^{F} + B^{F}, \left(AB\right)^{F} = B^{F}A^{F}.$$

**Proposition 10***If A and B are persymmetric matrices, then i*) *aA* + *bB*, *with a*, *b* ∈ ℝ, *ii*) *A*− 1, *if A*− 1 *exists, and iii*) *AT*, *are also persymmetric.*

\*\*Proof.\*\*  $i) \ (aA + bB)^{\circ} = aA^{\circ} + bB^{\circ} = aA + bB.$  

(ii)  $I = A^{-1}$   $A$  implies  $A^{\circ}$ ( $A^{-1}$ ) $^{\circ}$   $= A$  ( $A^{-1}$ ) $^{\circ}$   $= I$ . Hence,  $(A^{-1})^{\circ} = A^{-1}$ .

(iii)  $(A^{\circ})^{\circ}$   $= (A^{\circ})^{\circ}$   $= A^{\circ}$ .

0

714 014 2 6 4 with spectrum , and 2 0 4 . 354 350

> 000111444 000111444 000111444 222000444 <sup>1</sup> <sup>222000444</sup> , and <sup>3</sup> 222000444 333555000 333555000 333555000

é ù ê ú

é ù é ù ê ú ê ú <sup>=</sup> L = ê ú ê ú ê ú ê ú ë û ë û

*B C*

1

*A*

*T*

=

*XCX*

**5. NIEDP for persymmetric matrices**

and the following properties are straightforward:

2

is positive generalized doubly stochastic matrix with the prescribed elementary divisors.

In this section, we consider the *NIEDP* for persymmetric matrices. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations, where the eigenvalues of the Gram matrix, which is symmetric and persymmetric, play an important role [22]. As the superscript *T*, in *AT*, denotes the transpose of *A*, the superscript *F*, in *AF*, denotes the flip-transpose of *A*, which flips *A* across its skew-diagonal. If *A* = (*aij*)*mn*, then *AF* = (*an* − *j* + 1,*m* − *<sup>i</sup>* + 1)*nm*. A matrix *A* is said to be *persymmetric* if *AF* = *A*, that is, if it is symmetric across its lower-left to upper-right diagonal. Let *J* be the *n* × *n* matrix with ones along its skew-diagonal and zeroes elsewhere, that is, *J* = [**e***n*|**e***<sup>n</sup>* <sup>−</sup> 1| ⋯ |**e**1].

<sup>2</sup> ,, , *T F F T J J J J I A JA J* == = =

é ù ê ú = + ê ú ê ú ë û

*A A XCX A*

3

ë û

*T*

Finally

106 Applied Linear Algebra in Action

Then

In this section, we give sufficient conditions for the existence of nonnegative persymmetric matrices with prescribed elementary divisors. Our result generates an algorithmic procedure to compute a solution matrix. We also show that companion matrices are similar to persym‐ metric ones. As a consequence, we show that any realizable list of complex numbers *Λ* = {*λ*1, …, *λn*}, with Re*λ<sup>i</sup>* ≤ 0, *i* = 2, …, *n*, is in particular realizable by a nonnegative persymmetric matrix.

In [23], the authors develop the following persymmetric version of Rado's Theorem 3, which allow us to obtain sufficient conditions for the existence of a persymmetric nonnegative matrix with prescribed spectrum.

**Theorem 11** [23] *Let A be an n* × *n persymmetric matrix with spectrum*

*Λ* = {*λ*1, *λ*2, …, *λn*}, *and for some r* ≤ *n*, *let* {**x**1, **x**2, …, **x***r*} *be a set of eigenvectors of A corresponding to λ*1, …, *λr*, *respectively. Let X be the n* × *r matrix with i* − *th column* **x***<sup>i</sup> and rank*(*X*) = *r*. *Let Ω* = *diag*{*λ*1, …, *λr*}, *and let C be an r* × *r persymmetric matrix. Then, the matrix A* + *XCXF is persymmetric with eigenvalues μ*1, *μ*2, …, *μr*, *λ<sup>r</sup>* + 1, …, *λn*, *where μ*1, *μ*2, …, *μr are eigenvalues of B* = *Ω* + *CXF X*.

Let *Λ* = {*λ*1, *λ*2, …, *λn*} be a list of complex numbers, which can be partitioned as

$$\Lambda = \Lambda\_0 \bigcup \Lambda\_1 \bigcup \dots \bigcup \Lambda\_{\frac{p\_0}{2}} \bigcup \Lambda\_{\frac{p\_0}{2}} \cup \dots \cup \Lambda\_1, \text{for even } p\_0,$$

$$\text{with } \qquad \Lambda\_0 = \left\{ \lambda\_{01}, \lambda\_{02}, \dots, \lambda\_{0p\_0} \right\}, \quad \lambda\_{01} = \lambda\_1,$$

$$\Lambda\_k = \left\{ \lambda\_{11}, \lambda\_{12}, \dots, \lambda\_{kp\_k} \right\}, \ k = 1, 2, \dots, \frac{p\_0}{2},$$

where some of the lists *Λk* can be empty. For each list *Λk*, we associate the list

$$
\Gamma\_k = \left\{ \alpha \mathfrak{o}\_k, \mathfrak{X}\_{\mathbb{k}1}, \mathfrak{X}\_{\mathbb{k}2}, \dots, \mathfrak{X}\_{\mathbb{k}p\_1} \right\}, 0 \le \alpha \mathfrak{o}\_k \le \mathfrak{X}\_{\mathbb{k}}, \tag{10}
$$

which is realizable by a (*pk* + 1) × (*pk* + 1) nonnegative matrix *Ak*. In particular, *Ak* can be chosen as *Ak* ∈CS*ω<sup>k</sup>* .

Let the *n* × *p*0 matrix

$$X = \begin{bmatrix} \mathbf{x}\_1 \\ & \ddots \\ & & \mathbf{x}\_{\frac{p\_0}{2}} \\ & & & \mathbf{y}\_{\frac{p\_0}{2}} \\ & & & & \mathbf{y}\_{\frac{p\_0}{2}} \\ & & & & & \ddots \\ & & & & & & \mathbf{y}\_i \end{bmatrix}, \text{ for even } p\_0. \tag{11}$$

where, **x***k* = **e** = [1, 1, …, 1]*<sup>T</sup>* and **y***<sup>k</sup>* <sup>=</sup> *yk* 1, …, *yk* ( *pk* +1) *<sup>T</sup>* <sup>≥</sup><sup>0</sup> are, respectively, the Perron eigenvec‐ tors of *Ak* and *Ak <sup>F</sup>* , *<sup>k</sup>* =1, 2, …, *<sup>p</sup>*<sup>0</sup> <sup>2</sup> . It was shown in [23] that in the case of even *p*0, *XF X* is diagonal persymmetric nonnegative matrix with positive main diagonal. To apply Theorem 11, we need to find a *p*0 × *p*0 persymmetric nonnegative matrix *C*, where *B* = *Ω* + *CXF X* is persymmetric nonnegative with eigenvalues *μ*1, *<sup>μ</sup>*2, …, *μp*<sup>0</sup> (the new eigenvalues) and diagonal entries *ω*1, …, *ω <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *<sup>ω</sup>*1 (the former eigenvalues), and *<sup>Ω</sup>* <sup>=</sup>*diag*{*ω*1, …, *<sup>ω</sup> <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *<sup>ω</sup>*1}. Thus, since *XF X* is nonsingular, we may compute *C* = (*B* − *Ω*)(*XF X*) <sup>−</sup> 1. However, although both matrices, (*B* − *Ω*) and (*XF X*) − 1 are persymmetric, their product need not to be persymmetric. It was shown in [23, Lemma 3.1] that we may transform the matrix *X* into a matrix *X*˜ in such a way that *X*˜*<sup>F</sup> <sup>X</sup>*˜ <sup>=</sup>*sI*, *<sup>s</sup>* > 0, with the columns of *X*˜ still being eigenvectors of *A*. Now we state the main result of this section, which was proven in [10]. We only consider the case of even *p*0. The odd case is similar.

**Theorem 12** [10]*Let Λ* = {*λ*1, *λ*2, …, *λn*} *be a list of complex numbers with <sup>Λ</sup>* <sup>=</sup>*<sup>Λ</sup>*¯, *<sup>λ</sup>*<sup>1</sup> <sup>≥</sup> |*λ<sup>i</sup>* |, *i* = 2, 3, …, *n*, *and* ∑ *i*=1 *n λ<sup>i</sup>* ≥0. *Suppose there exists a partition of Λ as in (9), such that the following conditions are*

*<sup>i</sup>*) *For eachk* =1, 2, …, *<sup>p</sup>*<sup>0</sup> <sup>2</sup> ,*there exists a nonnegative matrix with spectrum*

$$
\Gamma\_k = \left( \alpha\_k, \mathcal{A}\_{k1}, \mathcal{A}\_{k2}, \dots, \mathcal{A}\_{kp\_k} \right), \\
0 \le \alpha\_k \le \mathcal{A}\_1,
$$

*and with certain prescribed elementary divisors.*

*satisfied:*

*ii*) *There exists a persymmetric nonnegative matrix of order p*0, *with spectrum Λ*0, *diagonal entries ω*1, *ω*2, …, *ω <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *ω*2, *ω*1,*and certain prescribed elementary divisors.*

*Then Λ is realizable by an n* × *n persymmetric nonnegative matrix with the prescribed elementary divisors.*

**Proof**. From *i*), let *Ak* be a nonnegative matrix with spectrum *Γk*, and certain prescribed elementary divisors, *<sup>k</sup>* =1, 2, …, *<sup>p</sup>*<sup>0</sup> <sup>2</sup> . We may assume that *Ak* ∈CS*ω<sup>k</sup>* . Then *Ak* **e** = *ω<sup>k</sup>* **e**, with **e** = (1, 1, …, 1)*<sup>T</sup>*. The matrix

$$A = A\_1 \oplus A\_2 \oplus \cdots \oplus A\_{\frac{p\_0}{2}} \oplus A\_{\frac{p\_0}{2}}^F \oplus \cdots \oplus A\_2^F \oplus A\_1^F$$

is persymmetric nonnegative with spectrum *Γ* =*Γ*<sup>1</sup> ∪⋯∪*Γ <sup>p</sup>* 0 2 ∪*Γ <sup>p</sup>* 0 2 ∪⋯∪*Γ*1, and the prescribed elementary divisors associated to the lists *Λk*. Let *X* (or *X*˜) be the matrix in (11) such that *<sup>X</sup> <sup>F</sup> <sup>X</sup>* <sup>=</sup>*sI <sup>p</sup>*<sup>0</sup> , *<sup>s</sup>* > 0 (or *X*˜*<sup>F</sup> <sup>X</sup>*˜ <sup>=</sup>*sI <sup>p</sup>*<sup>0</sup> , *<sup>s</sup>* > 0). Now, from *ii*) let *<sup>B</sup>* <sup>=</sup>*<sup>Ω</sup>* <sup>+</sup> *C X <sup>F</sup> <sup>X</sup>* <sup>=</sup>*<sup>Ω</sup>* <sup>+</sup> *sCI <sup>p</sup>*<sup>0</sup> be the *p*0 × *p*<sup>0</sup> persymmetric nonnegative matrix with spectrum *Λ*0 and diagonal entries *ω*1, …, *ω <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *<sup>ω</sup>*1. Then *<sup>C</sup>* <sup>=</sup> <sup>1</sup> *<sup>s</sup>* (*B* −*Ω*) is persymmetric nonnegative, where *<sup>Ω</sup>* <sup>=</sup>*diag*{*ω*1, …, *<sup>ω</sup> <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *<sup>ω</sup>*1}. Thus, from Theorem 11, *M* = *A* + *XCXF* is persymmetric with spectrum *Λ*, and with the prescribed elementary divisors. Since *A*, *X*, and *C* are nonnegative, then *M* is also nonnegative. ■

In [10], the authors show the following result:

**Theorem 13** [10] *Let*

Let the *n* × *p*0 matrix

108 Applied Linear Algebra in Action

tors of *Ak* and *Ak*

0 2 , *ω <sup>p</sup>* 0 2

matrices, (*B* − *Ω*) and (*XF X*)

The odd case is similar.

*<sup>i</sup>*) *For eachk* =1, 2, …, *<sup>p</sup>*<sup>0</sup>

*ω*1, …, *ω <sup>p</sup>*

*and* ∑ *i*=1

*satisfied:*

*ω*1, *ω*2, …, *ω <sup>p</sup>*

*divisors.*

0 2 , *ω <sup>p</sup>* 0 2

*n*

0

é ù ê ú

ë û

*p*

**x**

2

1

O

since *XF X* is nonsingular, we may compute *C* = (*B* − *Ω*)(*XF X*)

**x**

=

*<sup>F</sup>* , *<sup>k</sup>* =1, 2, …, *<sup>p</sup>*<sup>0</sup>

nonnegative with eigenvalues *μ*1, *<sup>μ</sup>*2, …, *μp*<sup>0</sup>

*and with certain prescribed elementary divisors.*

0

*p X p*

**y**

2

1

**y**

O

where, **x***k* = **e** = [1, 1, …, 1]*<sup>T</sup>* and **y***<sup>k</sup>* <sup>=</sup> *yk* 1, …, *yk* ( *pk* +1) *<sup>T</sup>* <sup>≥</sup><sup>0</sup> are, respectively, the Perron eigenvec‐

persymmetric nonnegative matrix with positive main diagonal. To apply Theorem 11, we need to find a *p*0 × *p*0 persymmetric nonnegative matrix *C*, where *B* = *Ω* + *CXF X* is persymmetric

, …, *<sup>ω</sup>*1 (the former eigenvalues), and *<sup>Ω</sup>* <sup>=</sup>*diag*{*ω*1, …, *<sup>ω</sup> <sup>p</sup>*

It was shown in [23, Lemma 3.1] that we may transform the matrix *X* into a matrix *X*˜ in such a way that *X*˜*<sup>F</sup> <sup>X</sup>*˜ <sup>=</sup>*sI*, *<sup>s</sup>* > 0, with the columns of *X*˜ still being eigenvectors of *A*. Now we state the main result of this section, which was proven in [10]. We only consider the case of even *p*0.

*λ<sup>i</sup>* ≥0. *Suppose there exists a partition of Λ as in (9), such that the following conditions are*

<sup>2</sup> ,*there exists a nonnegative matrix with spectrum*

{ 1 2 } <sup>1</sup> , , , , ,0 , *<sup>k</sup>* G= ¼ £ £ *k k k k kp*

*ii*) *There exists a persymmetric nonnegative matrix of order p*0, *with spectrum Λ*0, *diagonal entries*

*Then Λ is realizable by an n* × *n persymmetric nonnegative matrix with the prescribed elementary*

, …, *ω*2, *ω*1,*and certain prescribed elementary divisors.*

 l  w l*k*

**Theorem 12** [10]*Let Λ* = {*λ*1, *λ*2, …, *λn*} *be a list of complex numbers with <sup>Λ</sup>* <sup>=</sup>*<sup>Λ</sup>*¯, *<sup>λ</sup>*<sup>1</sup> <sup>≥</sup> |*λ<sup>i</sup>*

wl l 0

<sup>2</sup> . It was shown in [23] that in the case of even *p*0, *XF X* is diagonal

− 1 are persymmetric, their product need not to be persymmetric.

(the new eigenvalues) and diagonal entries

0 2 , *ω <sup>p</sup>* 0 2 ,

<sup>−</sup> 1. However, although both

…, *<sup>ω</sup>*1}. Thus,


(11)

, for even .

$$C = \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \ddots & \mathbf{0} \\ \ddots & \ddots & \ddots & \ddots & \vdots \\ \mathbf{0} & \ddots & \ddots & \ddots & 1 \\ \mathbf{c}\_{n-1} & \cdots & \mathbf{c}\_{2} & \mathbf{c}\_{1} & \mathbf{c}\_{0} \end{bmatrix}$$

*be the companion matrix of the polynomialp*(*x*)= *x <sup>n</sup>* + ∑ *k*=1 *n ck* <sup>−</sup>1*<sup>x</sup> <sup>n</sup>*−*<sup>k</sup> . Then*C*is similar to a persymmetric matrix*P*. In particular, if*C*is nonnegative,*P*is also nonnegative.*

In [21], Laffey and Šmigoc solve the NIEP in the left half plane. That is, they give necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed complex spectrum *Λ* = {*λ*1, *λ*2, …, *λn*} satisfying R*eλ<sup>i</sup>* ≤ 0, *i* = 2, …, *n*. Here we show that a realizable list of complex numbers, with R*eλ<sup>i</sup>* ≤ 0, *i* = 2, …, *n*, is in particular realizable by a nonnegative persymmetric matrix.

**Theorem 14** [10] *Let Λ* = {*λ*1, *λ*2, …, *λn*}, *with* R*eλ<sup>i</sup>* ≤ 0, *i* = 2, …, *n*, *be a realizable list of complex numbers. Then Λ is also realizable by a nonnegative persymmetric matrix.*

**Proof**. If *Λ* is realizable, then from the result in [21], *Λ* is the spectrum of a matrix of the form C + *αI*, where C is a nonnegative companion matrix with *tr*(C) =0. Then from Theorem 13, C is similar to a nonnegative persymmetric matrix P with eigenvalues *λ*<sup>1</sup> − *α*, *λ*<sup>2</sup> − *α*, …, *λ<sup>n</sup>* − *α*. Hence P + *αI* is nonnegative persymmetric with spectrum *Λ*.

To apply Theorem 12, we need to know conditions under which there exists a *p*0 × *p*0 persym‐ metric nonnegative matrix with spectrum *Λ*0 and diagonal entries *ω*1, *ω*2, …, *ω <sup>p</sup>* 0 2 , *ω <sup>p</sup>* 0 2 , …, *ω*2, *ω*1. Following a result due to Farahat and Ledermann [24, Theorem 2.1], we have

#### **Theorem 15***[24] Let Λ* = {*λ*1, *λ*2, …, *λn*} *be a list of complex numbers and let* {*a*1, *a*2, …, *an*} *be a list of nonnegative real numbers such that* ∑ *i*=1 *n ai* <sup>=</sup>∑ *i*=1 *n λi* . *Let p*(*x*) = (*x* − *λ*1)(*x* − *λ*2) ⋯ (*x* − *λn*) *and*

$$
\mu\_0 = 1, \mu\_1 = (\infty - a\_1), \dots, \mu\_n = (\infty - a\_1)(\infty - a\_2)\cdots(\infty - a\_n),
$$

with

$$p\left(\mathbf{x}\right) = \mu\_n + k\_1\mu\_{n-1} + \dots + k\_{n-1}\mu\_1 + k\_n. \tag{12}$$

*If k*1, *k*2, …, *kn are all nonpositive, then there exists an n* × *n nonnegative matrix, with spectrum Λ and with diagonal entries a*1, *a*2, …, *an*.

**Proof**. The polynomials *μ*0, *μ*1, …, *μn* constitute a basis in the space of polynomials of degree less than or equal to *n*. Equating the coefficients of *xn* − 1 in both sides in (12), we obtain *k*1 = 0. Let the matrix


Expanding the determinant of *xI* − *A* with respect to the last row it, follows that *A* has charac‐ teristic polynomial *p*(*x*) and therefore *A* has spectrum *Λ*. Besides, if *k*1, …, *kn* are all nonpositive, *A* is nonnegative.

**Example 6***Let Λ* = {9, − 3, − 3, − 3}. *First we compute a persymmetric nonnegative matrix* P *with spectrum Λ and with elementary divisors* (*λ* − 9), (*λ* + 3)3 . *Then p*(*x*) = (*λ* − 9) (*λ* + 3)3 = *λ*<sup>4</sup> − 54*λ*<sup>2</sup> − 216*λ* − 243, *and the corresponding matrices* C *(the companion of p*(*x*)*) and* P *(obtained from Theorem 13) are, respectively,*

$$C = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 243 & 216 & 54 & 0 \end{bmatrix} \rightarrow P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 27 & 0 & 1 & 0 \\ 108 & 0 & 0 & 1 \\ 972 & 108 & 27 & 0 \end{bmatrix}.$$

*Next, we compute a persymmetric matrix*P*with spectrum Λ and linear elementary divisors: Then we take the partition*

$$
\Lambda\_0 = \left\{ \mathfrak{P}, -\mathfrak{Z} \right\}, \\
\Lambda\_1 = \Lambda\_2 = \left\{ -\mathfrak{Z} \right\} \\
\text{with } \Gamma\_1 = \Gamma\_2 = \left\{ \mathfrak{Z}, -\mathfrak{Z} \right\}.
$$

*We apply Theorem 12 with*

similar to a nonnegative persymmetric matrix P with eigenvalues *λ*<sup>1</sup> − *α*, *λ*<sup>2</sup> − *α*, …, *λ<sup>n</sup>* − *α*.

To apply Theorem 12, we need to know conditions under which there exists a *p*0 × *p*0 persym‐ metric nonnegative matrix with spectrum *Λ*0 and diagonal entries

**Theorem 15***[24] Let Λ* = {*λ*1, *λ*2, …, *λn*} *be a list of complex numbers and let* {*a*1, *a*2, …, *an*} *be a list of*

01 1 ( ) ( )( ) ( ) 1 2 1, , , , *n n*

 m

 m= =- =- - - *xa xa xa xa* L L

( ) 1 1 1 1 . *nn n n px k k k* = + ++ +

*If k*1, *k*2, …, *kn are all nonpositive, then there exists an n* × *n nonnegative matrix, with spectrum Λ and*

**Proof**. The polynomials *μ*0, *μ*1, …, *μn* constitute a basis in the space of polynomials of degree less than or equal to *n*. Equating the coefficients of *xn* − 1 in both sides in (12), we obtain *k*1 = 0.

mm

1

*a*

*A*

*spectrum Λ and with elementary divisors* (*λ* − 9), (*λ* + 3)3

*(obtained from Theorem 13) are, respectively,*

=

2

OOO O OO O L

0 1

*a*

3 2

*n n*

*k k ka*

ë û - --

Expanding the determinant of *xI* − *A* with respect to the last row it, follows that *A* has charac‐ teristic polynomial *p*(*x*) and therefore *A* has spectrum *Λ*. Besides, if *k*1, …, *kn* are all nonpositive,

**Example 6***Let Λ* = {9, − 3, − 3, − 3}. *First we compute a persymmetric nonnegative matrix* P *with*

= *λ*<sup>4</sup> − 54*λ*<sup>2</sup> − 216*λ* − 243, *and the corresponding matrices* C *(the companion of p*(*x*)*) and* P

0 1

L O M

0 .

10 0

é ù ê ú

, …, *ω*2, *ω*1. Following a result due to Farahat and Ledermann [24, Theorem

. *Let p*(*x*) = (*x* − *λ*1)(*x* − *λ*2) ⋯ (*x* − *λn*) *and*


. *Then p*(*x*) = (*λ* − 9)

Hence P + *αI* is nonnegative persymmetric with spectrum *Λ*.

*i*=1

*n ai* <sup>=</sup>∑ *i*=1

*n λi*

*ω*1, *ω*2, …, *ω <sup>p</sup>*

110 Applied Linear Algebra in Action

2.1], we have

with

Let the matrix

*A* is nonnegative.

(*λ* + 3)3

0 2 , *ω <sup>p</sup>* 0 2

*nonnegative real numbers such that* ∑

*with diagonal entries a*1, *a*2, …, *an*.

mm

$$A = \begin{bmatrix} 0 & 3 & 0 & 0 \\ 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 \end{bmatrix}, \quad X = \begin{bmatrix} 1 \\ \overline{\sqrt{2}} & 0 \\ 1 \\ \overline{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \\ \overline{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 36 & 3 \end{bmatrix},$$

*to obtain the persymmetric nonnegative matrix A* + *XCXF with the required properties. Finally, we compute a persymmetric nonnegative matrix* P *with spectrum Λ and with elementary divisors* (*λ* − 9), (*λ* + 3)<sup>2</sup> , (*λ* + 3). *In this case, we take the partition*

$$
\Lambda\_0 = \left\{ \mathbf{9}, -\mathbf{3}, -\mathbf{3} \right\}, \Lambda\_1 = \Lambda\_3 = \mathcal{O}, \Lambda\_2 = \left\{ -\mathbf{3} \right\}.
$$

$$
\Gamma\_1 = \Gamma\_3 = \left\{ \mathbf{0} \right\}, \Gamma\_2 = \left\{ \mathbf{3}, -\mathbf{3} \right\}
$$

$$
A\_1 = A\_3 = \begin{bmatrix} \mathbf{0} \end{bmatrix}, \ A\_2 = \begin{bmatrix} \mathbf{0} & \mathbf{3} \\ \mathbf{3} & \mathbf{0} \end{bmatrix}.
$$

*We compute the matrix B*, *with spectrum Λ*0*and diagonal entries* 0, 3, 0 *(in that order), from Theorem* 15, *case n* = 3:

$$B = \begin{bmatrix} 0 & 1 & 0 \\ \hline 45 & 3 & 1 \\ 2 & 3 & 1 \\ 81 & \frac{45}{2} & 0 \end{bmatrix} \\
\text{with } J(B) = \begin{bmatrix} 9 & 0 & 0 \\ 0 & -3 & 1 \\ 0 & 0 & -3 \end{bmatrix}$$

*Then,*

$$\begin{split} \boldsymbol{A} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 3 & 0 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \\ \frac{45}{2} & 0 & 1 \\ 81 & \frac{45}{2} & 0 \\ 81 & \frac{45}{2} & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 0 & \frac{1}{2}\sqrt{2} & \frac{1}{2}\sqrt{2} & 0 \\ \frac{45}{4}\sqrt{2} & 0 & 3 & \frac{1}{2}\sqrt{2} \\ \frac{45}{4}\sqrt{2} & 3 & 0 & \frac{1}{2}\sqrt{2} \\ \frac{45}{4}\sqrt{2} & \frac{45}{4}\sqrt{2} & 0 \end{bmatrix}, \quad J(A) = \begin{bmatrix} 9 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & -3 \end{bmatrix}. \end{split}$$
