不可約非負矩陣相似至正矩陣的問題探討

國

立

交

通

大

學

應用數學系

碩

士

論

文

不可約非負矩陣相似至正矩陣的問題探討

Irreducible nonnegative matrices that are similar to positive

matrices

研 究 生：王信元

指導教授：王國仲 教授

不可約非負矩陣相似至正矩陣的問題探討

Irreducible nonnegative matrices that are similar to positive matrices

研 究 生：王信元 Student：Hsin-Yuan Wang

指導教授：王國仲 Advisor：Kuo-Zhong Wang

國 立 交 通 大 學

應用數學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Master

in

Applied Mathematics

July 2011

不可約非負矩陣相似至正矩陣的問題探討

研究生：王信元 指導教授：王國仲

國立交通大學應用數學系(研究所)碩士班

摘 要

我們所要探討的是不可約非負矩陣相似至正矩陣的問題，我們

將探討二個主要問題。一是對於 n 階不可約非負矩陣且有 n+2 個零元

素和對角線元素皆相異的條件之下，會相似至正矩陣。二是對於 n 階

不可約非負矩陣且有 n+2 個非零元素的條件之下，會相似至正矩陣。

Irreducible nonnegative matrices that are similar to positive matrices

student：Hsin-Yuan Wang Advisors：Kuo-Zhong Wang

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

We study the questions in which the irreducible nonnegative matrices are similar

to positive matrices.We will solve the problems for n×n irreducible nonnegative

matrices with exactly n+2 zeros, and all the entries of diagonal are distinct, and for n×

n irreducible nonnegative matrices with exactly n+2 nonzero elements are similar to

positive matrices.

誌 謝

時光匆匆，在研究所的求學階段也即將告一段落。首先我要感謝我的指

導教授王國仲老師，在我學習上遇到困難時，總是能夠適時地給我一些指點，老

師做研究的嚴格態度，相信對我將來會有極大的幫助，同時也要感謝幫我口試的

兩位老師，林敏雄教授和蔣俊岳教授，感謝他們在口試時給我的寶貴意見，讓我

的論文能更完善。

最後我想要感謝我的父母，在我感到無力有他們的鼓勵，讓我倍感溫馨，

也更有動力向前邁進。還有我的研究所師長，同學，及好友們，願將我的成果與

你們一同分享。

目

錄

中文摘要

………

i

英文摘要

………

ii

誌謝

………

iii

目錄

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iv

1.Introduction

………

1

2.Preliminaries

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2

3.Matrix with exactly n + 2 zeros

………

4

4. Matrix with exactly n + 2 nonzero entries ………

8

1

Introduction

For a matrix A = [aij]ni,j=1, we say that A is nonnegative is all its entries

aij are nonnegative. We say that A is positive (A > 0) if all its entries aij

are positive. A nonnegative matrix A is called reducible matrix if there is a permutation matrix Q such that

QTAQ =  B C 0 D  ,

where B, C are square matrices, and it is called irreducible if it is not reducible. We are interested in irreducible nonnegative matrices that are similar to positive matrices. This question was first studied in the paper of Borobia and Moro[1], where they obtained several results on the problem. Recently, in [2], Laffey, Loewy, and ˇSmigoc proved two main theorems:

LLS 1. Let A be an n × n, n ≥ 4, irreducible nonnegative matrix with exactly n zero elements. Then either A has trace zero or it is similar to a positive matrix.

LLS 2. Every irreducible n × n nonnegative matrix A with positive trace and exactly n + 1 zeros is similar to a positive matrix for all n ≥ 3.

In this paper, we consider the following problems:

Question 1. For n ≥ 4, is every irreducible n × n nonnegative matrix A with positive trace and exactly n + 2 zeros similar to a positive matrix ?

Question 2. Is every irreducible n × n nonnegative matrix A with positive trace and exactly n + 1 or n + 2 nonzeros similar to a positive matrix ? In Question 2, if “exactly n + 1 or n + 2 nonzeros00 replaces by “exactly n+3 nonzeros00, then the answer is false.

Counterexample [1, Theorem 5]: For a ∈ (0, 1),   a 0 1 − a 1 − a a 0 0 1 − a a 

 is not similar to a positive matrix. In section 3, we prove that for n ≥ 4 and n 6= 5, if A is an n × n irreducible nonnegative matrix with exactly n + 2 zeros and distinct diagonal entries, then A is similar to a positive matrix. In section 4, we solve Question 2.

2

Priliminaries

In the section, we list some Lemma, which will be used in the proofs of our main result. The following lemmas appeared in [2, Corollary 3.4, and Lemma 3.5].

Lemma 2.1 Let A be an irreducible matrix with a positive row or column.Then A is similar to a positive matrix.

Lemma 2.2 Let A =  A11 A12 A21 A22 

be a nonnegative matrix such that A11 is similar to a positive matrix, A12> 0,

and A21> 0. Then A is similar to a positive matrix.

Let P be a nonnegative pattern matrix (i.e. a matrix whose entries are either zeros or positive entries), and let Pi, Pj(Pi, Pj) be two columns(rows) of P . We

say that Pj dominates Pi(Pj dominates Pi), and we write Pj ≥ Pi(Pj ≥ Pi),

if Pj(Pj) has positive entries in at least the same positions as Pi(Pi). We denote

by Pi + Pj(Pi+ Pj) the pattern column with stars in all positions expect in

those where both Pi and Pj(Pi and Pj) have zeros.

The following two lemmas were proved in [1,Theorems 2 and 4]. We need them to derive our main results. For the sake of completion, we list their proofs below.

Lemma 2.3 Let P be an irreducible pattern matrix with rows P1, . . . , Pn and

columns P1, . . . , Pn, and suppose Pi≤ Pj_(P

i≤ Pj) for a certain pair of indices

i, j.Then any nonnegative matrix with pattern P is similar to a nonnegative irreducible matrix with pattern Q,where Qi = Pi+ Pj and Qk = Pk for k 6= i

(Qi = Pi+ Pj and Qk= Pk for k 6= i).

Proof. Suppose that i = 1, j = 2, and P1 ≤ P2_{(the proof in the row case is}

completely analogous). We consider the n × n matrix (2.1) X =  Y 0 0 In−2  , where Y =  1  0 1 

and  is positive. Let A be any nonnegative matrix with pattern P . We partition A =



A11 A12

A21 A22

conformally with (2.1). Then XAX−1 =  Y A11Y−1 Y A12 A21Y−1 A22  . The block A21Y−1 =    a31 a32− a31 .. . ... an1 an2− an1   

has, for  small enough, the same pattern as A21, since ak2 can only be zero if

ak1 is zero as well. Note that A22does not change, so the row 3,. . . , n keep the

same pattern as before the change of variable. The same applies to the second row, since Y A21Y−1=  a11+ a21 a12+ (a22− a11− 2a21) a21 a22− a21  , Y A12=  a13+ a23 · · · a1n+ a2n a23 · · · a2n  .

Finally, the first row can only increase its pattern, and this will only happen in those positions where a1k= 0, a2k > 0. In other words, the pattern of the first

row becomes P1+ P2 and the rest remain invariant.

Lemma 2.4 Let A = [aij]ni,j=1 be a nonnegative irreducible matrix with aij =

0, i 6= j, and suppose that akj > 0 for k 6= i (aik> 0 for k 6= j).If aii< ajj(aii>

ajj).Then A is similar to a positive matrix.

Proof. Suppose again that i = 1, j = 2, and satisfies ak2> 0 for k 6= 2(the proof

with i = 1, j = 2, and satisfies a1k > 0 for k 6= 1 is carried out analogously).

Hence, A is the form

A =        a11 0 · · · a22 a32 .. . an2        .

with a12= 0, ak2 > 0 for k 6= 1, and a11< a22. We consider again the change of

variables (2.1). One can check, as in the proof of Lemma 2.4, that for  small enough the matrix XAX−1 is nonnegative and its pattern strictly dominates

the pattern of A. Thus, XAX−1 is irreducible, and its second column        (a22− a11− a21) a22− a21 a32− a31 .. . an2− an1        .

is positive. Hence, according to Lemma 2.2, XAX−1 is similar to a positive matrix.

3

Matrix with exactly n + 2 zeros

Proposition 3.1 For n ≥ 6, let A = [aij]n_i,j=1 be an irreducible n × n

nonneg-ative matrix with exactly n + 2 zeros and let the diagonal entries be all distinct . If an1= an2 = an3= 0, then A is similar to a positive matrix.

Proof. First, suppose that zeros in rows 1, 2, 3 does not appear in the same column. Since there is a column in the first three columns with exactly a zero, we may assume that it is the third column. Then A3 ≥ A1_{, A}3 _{≥ A}2_{. By}

Lemma 2.3, we know that A is similar to a nonnegative irreducible matrix Q, where Q1 = A1+ A3, Q2 = A2+ A3, and Qk = Ak, k 6= 1, 2. Moreover, Q has

exactly k zeros, where k < n + 2. Hence, we apply LLS 1 and LLS 2 to show that Q is similar to a positive matrix, and this implies that A is similar to a positive matrix.

Next, we consider the situation when zeros in rows 1, 2, 3 appear in the same column.

(1) If column n has only a zero, we may assume that a14 = a24 = a34 = 0.

Then A is permutationally similar to the form A =  A11 A12 A21 A22  , where A11 is a k × k matrix with the pattern

            + + + 0 + · · · + + + + 0 + · · · + + + + 0 + · · · + + + + + 0 . .. ... .. . ... ... ... . .. ... + + + + + · · · + 0 0 0 0 + · · · +             ,

A22is an (n−k)×(n−k) matrix with n−k zeros, and A12, A21are positive

matrices. Since a(k−1)k = 0 with aik > 0, i 6= k − 1 and a(k−1)j > 0, j 6= k,

by Lemma 2.4, we have A11 is similar to a positive matrix. By Lemma

2.2, this implies that A is similar to a positive matrix.

(2) Suppose that a1n= a2n = a3n= 0. Then A is permutationally similar to

the form A =  A11 A12 A21 A22  , where A11 is a 4 × 4 matrix with the pattern

    + + + 0 + + + 0 + + + 0 0 0 0 +    

A22 is an (n − 4) × (n − 4) matrix with n − 4 zeros, and A12, A21 are

positive matrices. If n ≥ 8, then A22 is irreducible with nonzero trace.

By LLS 1, A22 is similar to a positive matrix. Hence, by Lemma 2.2, A

is similar to a positive matrix.

For 6 ≤ n < 8, there is an aik = 0, 4 < i, k ≤ n and i 6= k, satisfying

aij > 0, j 6= k and ajk > 0, j 6= i. By Lemma 2.4, A is similar to a

positive matrix.

Theorem 3.2 For n ≥ 4 and n 6= 5, let A = [aij]ni,j=1 be an irreducible n × n

nonnegative matrix with exactly n + 2 zeros. If the diagonal entries are distinct, then A is similar to a positive matrix.

Proof. By Lemma 2.1, we may assume that A has at least a zero in every row and column. It only need to consider that row n has three zeros or two zeros.

Case 1. Row n has three zeros:

From Proposition 3.1, we only need to consider the case an1 = an2= ann = 0. If

there exist ai1j = ai2j = ai3j = 0 and ajj > 0. Then we can replace A by A

T _and

use the arguments from Proposition 3.1 to finish the proof. Next, if zeros in rows 1, 2, n appear in the same column and a zero of them is on the main diagonal. Since the diagonal entries are distinct, we have a1n = a2n = ann = 0.Then

A1 ≥ An_{. By Lemma 2.3, A is similar to a nonnegative irreducible matrix Q,}

where Qn= A1+ Anand Qk= Ak for k 6= n. Moreover, Q has exactly k zeros,

where k < n + 2. Thus A is similar to a positive matrix. Finally, We consider the situation when zeros in rows 1, 2, n do not appear in the same column. In the situation, if n ≥ 6, then there is an aij = 0, 3 ≤ j ≤ 5, i 6= j such that

akj > 0 for k 6= i and ail> 0 for l 6= j. By Lemma 2.4, A is similar to a positive

matrix. For n = 4, applying Lemma 2.4, we only need to consider the pattern:     + + 0 + + + 0 + + + + 0 0 0 + 0     .

By Lemma 2.3, we know that it is similar to a nonnegative irreducible matrix with exactly k zeros, where k < n + 2. Hence it is similar to a positive matrix. Case 2. Two zeros in row n are off the main diagonal. We may assume that an1 = an2= 0:

1. There are two additional zeros in the first two columns:

If n ≥ 6, then there is an aij = 0, 3 ≤ j ≤ 5, i 6= j such that akj > 0

for k 6= i and ail > 0 for l 6= j. By Lemma 2.4, A is similar to a

positive matrix. For n = 4, using Lemma 2.4, we only need to consider the following patterns:

    + + 0 0 0 + + + + 0 + + 0 0 + +     ,     + 0 + + + + 0 0 0 + + + 0 0 + +     .     0 + + + + + 0 0 + 0 + + 0 0 + +     ,     + 0 + + 0 + + + + + 0 0 0 0 + +     .

Four patterns above, by Lemma 2.3, we know that these are similar to nonnegative irreducible matrices with exactly k zeros, where k < n + 2. Hence these are similar to positive matrices.

2. There is an additional zero in the first two columns:

We may assume that column 1 has two zeros. First, we consider the situation when zeros in rows 1, 2 do not appear in the same column. Since A2 ≥ A1_{, by Lemma 2.3, A is similar to a nonnegative irreducible}

matrix Q, where Q1 = A1+ A2, Qk = Ak for k 6= 1. Then Q has exactly

m zeros, where m < n + 2. Hence, Q is similar to a positive matrix and this implies that A is similar to a positive matrix.

Next, we consider the situation when zeros in rows 1, 2 appear in the same column. If n ≥ 6, then there is an aij = 0, 3 ≤ j ≤ 5, i 6= j such

to a positive matrix. For n = 4, by Lemmas 2.3 and 2.4, we only need to consider the following patterns:

A(1) =     + + 0 + + + 0 0 0 + + + 0 0 + +     , A(2) =     + + 0 + 0 + 0 + + + + 0 0 0 + +     , A(3) =     + + + 0 0 + + 0 + + 0 + 0 0 + +     .

All of the above patterns are discussed in Proposition 3.3. 3. The first two columns are both a zero:

First we consider the situation when zeros in rows 1, 2 do not appear in the same column. Since A2 ≥ A1_{. By Lemma 2.3, A is similar to a}

nonnegative irreducible matrix Q such that Q has exactly k zeros, where k < n + 2. Hence A is similar to a positive matrix. Next, suppose that a1r = a2r = 0. If r = n, from Lemmas 2.2 and 2.4, our analysis reduces

to the situation: B =     + + 0 + + + 0 + 0 0 + + + + + 0     .

By LLS 2, B is similar to a positive matrix. If 3 ≤ r < n, we may assume that r = 3 and n ≥ 6. By the hypothesis, there is an aij = 0, 4 ≤ j < n,

3 ≤ i < n, i 6= j such that akj > 0 for k 6= i and ail > 0 for l 6= j. By

Lemma 2.4, A is similar to a positive.

Case 3. Row n has two zeros and a zero in row n is on the main diagonal: We may assume that an1 = ann = 0. Then there exist ai1j = ai2j = 0

for some 1 ≤ i1, i2 ≤ n and i1, i2 6= j. Replace A by AT. By Case 2, we

obtain the proof.

Proposition 3.3 For 1 ≤ i ≤ 3, A(i) is defined as in the proof of Theorem 3.2. Then A(i) is similar to a positive matrix.

Proof. Let Eij denote a (0, 1) matrix with element (i, j) equal to one and all

other elements equal to zero. We define

Fij() = I − Eij.

Clearly, F_ij−1() = Fij(−).

For i = 1, 2, 3, A(i) = [aij]4i,j=1, if a22 < a44, F42()A(i)F₄₂−1() is

nonneg-ative and has positive column 2 for all sufficiently small  > 0. If a22 > a44,

F42(−)A(i)F42−1(−) is nonnegative and has positive column 2 for all sufficiently

4

Matrix with exactly n+2 nonzero entries

Suppose that A is an n × n irreducible nonnegative matrix with tr(A) > 0. In this section, we show that A is similar to a positive matrix when A has exactly n + 1 or n + 2 nonzeros entries.

Proposition 4.1 Let A be an n×n irreducible nonnegative matrix with exactly n+2 positive elements and tr(A) > 0. Suppose that A is permutationally similar to a matrix of the form

                         0 + + 0 . .. ... . .. ... . .. ... + 0 + ⊕ + ⊕ + 0 . .. ... . .. 0 + 0                          ,

where the positions of positive diagonal entries are (i − 1, i − 1)th and (i, i)th, and we replace + of diagonal entries by ⊕. Then A is similar to a positive matrix.

Proof. Obviously, Ai−1≥ Ai−2_{. By Lemma 2.3, A is similar to a nonnegative}

irreducible matrix B with pattern

B =                          0 + + 0 . .. ... . .. ... . .. ... + + + + ⊕ + ⊕ + 0 . .. ... . .. 0 + 0                          .

Using similar step, we get that A is similar to a nonnegative irreducible matrix C with pattern C =                        + + + · · · + + + + + + + · · · + + + + + · · · + + + . .. ... ... .._. .._. + + + + + + + + ⊕ + ⊕ + 0 . .. ... . .. 0 0 + 0                        ,

by observing this pattern, we have C1 ≥ Cn_{, and then it is similar to a}

non-negative irreducible matrix P with pattern

P =                       + + + · · · + + + + + + + · · · + + + + + · · · + + + . .. ... ... .._. .._. + + + + + + + + ⊕ + ⊕ + 0 . .. ... + 0 + + + · · · + + + 0 · · · + +                      

Using similar step, by Lemma 2.3, P is similar to a nonnegative irreducible matrix with pattern

                      + + + · · · + + + + + + + · · · + + + + + · · · + + + . .. ... ... .._. .._. + + + + + + + + ⊕ + ⊕ + + .. . . .. + + + + · · · + + + 0 · · · + +                       ,

where it has positive column i − 1. By Lemma 2.1, it is similar to a positive matrix.

From the proof of Proposition 4.1, the diagonal entry of position (i, i) can be zero. Hence we have the following theorem.

Theorem 4.2 Let A be an n × n irreducible nonnegative matrix with exactly n + 1 positive elements and tr(A) > 0. Then A is similar to a positive matrix. Proof. From the hypothesis, A is permutationally similar to a matrix of the form               0 an+1 a1 0 . .. ... ai−2 ai−1 ai 0 ai+1 0 . .. ... an 0               ,

where a1, · · · , an, an+1> 0. From the proof of Proposition 4.1, we show that A

is similar to a positive matrix.

Proposition 4.3 Let A be an n×n irreducible nonnegative matrix with exactly n + 2 positive elements and tr(A) > 0. If A is permutationally similar to a

matrix of the form                          0 + + 0 . .. ... + ⊕ + 0 + . .. . .. 0 + ⊕ + 0 . .. ... . .. 0 + 0                          ,

where the positions of positive diagonal entries are (i, i)th and (i + k, i + k)th, and we replace + of diagonal entries by ⊕. Then A is similar to a positive matrix.

Proof. Using the method of Proposition 4.1 and Lemma 2.3, A is similar to the following pattern matrix

P ≡                          + + + + 0 + + + + + 0 . .. ... ... ... + ⊕ 0 + 0 + . .. .. . . .. 0 0 0 + ⊕ + 0 + 0 .. . ... . .. ... + 0 . .. 0 + + + + 0 + +                          .

Since Pi ≥ Pi+1, by Lemma 2.3, P is similar to the following pattern matrix Q ≡                          + + + + + + + + + + + . .. ... ... ... + ⊕ + + + + . .. .. . . .. 0 0 + ⊕ + + 0 .. . . .. ... + . .. 0 + + + + + + +                          .

By the discussion above, Q is similar to a nonnegative irreducible matrix with pattern                          + + + + + + + + + + + + + . .. ... ... ... .._. + ⊕ + + + + + + . .. + .. . . .. + 0 + ⊕ + + + 0 .. . ... . .. ... + . .. 0 + + + + + + + +                          .

It has positive column i + k − 1 and then A is similar to a positive matrix.

Proposition 4.4 Let A be an n×n irreducible nonnegative matrix with exactly n + 2 positive elements and tr(A) > 0. If A satisfies the following properties:

(i) The diagonal entries have a nonzero element.

(ii) There is a positive (i, j) position with i 6= j, and ith row and jth column have an additional positive element respectively.

Then A is similar to a positive matrix.

We need the following lemma to prove Proposition 4.4.

Lemma 4.5 Let A = [ai,j]ni,j=1 be an irreducible nonnegative matrix with

ex-actly n + 1 positive elements and tr(A) = 0. If A satisfies (ii) of Proposition 4.4. Then A is permutationally similar to a matrix of the form

                 0 + + 0 + . .. . .. ... ⊕ + . .. . .. ... . .. ... 0 + 0                  . Proof. Obviously.

The proof of Proposition 4.4:

Here we replace the nonzero diagonal entry (k, k) and positive (i, j) position by notation ⊕. By Lemma 4.5, we only need to consider the following two cases. Case 1. ⊕ in the position (i, 1), where 3 ≤ i ≤ k:

A ≡                           0 0 0 + + 0 0 0 + 0 + . .. . .. ... ⊕ + . .. . .. 0 + ⊕ + 0 . .. ... . .. 0 0 + 0                           .

Since Ak ≥ Ak−1_{, by Lemma 2.3, A is similar to a nonnegative irreducible}

step, then A is similar to the following pattern C ≡                           0 0 0 · · · 0 0 + + + + · · · + + + + · · · + + + . .. ... ... ... ... . .. ... ⊕ + . .. ... ... . .. + + + ⊕ + 0 . .. ... . .. 0 0 + 0                           .

Since Ck ≥ Ck+1, by Lemma 2.3, C is similar to a nonnegative irreducible

matrix C0 with C0k+1 = Ck+ Ck+1 and C0r = Cr for r 6= k + 1. Repeat this step, then C is similar to the following pattern

D ≡                           0 0 0 · · · 0 0 0 + + + + · · · + + + + + + · · · + + + + + . .. ... ... ... ... ... ... . .. ... ⊕ + . .. ... ... ... . .. + + + + ⊕ + + + + . .. ... .._. . .. + + + +                           .

From Lemma 2.1, D is similar to a positive matrix. Case 2. ⊕ in the position (i, 1), where i > k:

A ≡                           0 0 0 + + 0 0 + 0 . .. ... + 0 + ⊕ + 0 . .. ... . .. ... ⊕ + 0 . .. ... 0 0 + 0                           .

Since Ak ≥ Ak−1_{, by Lemma 2.3, A is similar to a nonnegative irreducible}

matrix A0 with A0_k−1 = Ak+ Ak−1 and A0r = Ar for r 6= k − 1. Repeat this

step, A is similar to the following pattern

Q ≡                           0 0 0 · · · 0 0 0 + + + + · · · + + 0 + + · · · + + 0 . .. ... ... .._. .._. + + + 0 + ⊕ 0 + 0 . .. ... . .. ... ⊕ + 0 . .. ... 0 0 + 0                           .

Since Qk ≥ Qk+1, by Lemma 2.3, Q is similar to a nonnegative irreducible

step, Q is similar to the following pattern R ≡                           0 0 0 · · · 0 0 0 · · · 0 + + + + · · · + + + · · · + + + · · · + + + · · · + . .. ... ... .._. .._. .._. .._. + + + + · · · + + ⊕ + · · · + + + · · · + . .. ... ... . .. + ⊕ + 0 . .. ... 0 0 + 0                           .

Since Ri−1 ≥ R1_{, by Lemma 2.3, R is similar to a nonnegative irreducible}

matrix R0 with R0₁ = R1+ Ri−1 and R0r= Rr for r 6= 1. Repeat this step, R is

similar to the following pattern                           0 0 0 · · · 0 0 0 · · · + + + 0 0 · · · 0 0 + · · · + 0 + + · · · + + + · · · + . .. ... ... .._. .._. .._. .._. + + + + · · · + + ⊕ + · · · + + + · · · + . .. ... ... . .. + ⊕ + 0 .. . . .. ... + 0 0 + + 0                           ,

which is positive column i − 1. From Lemma 2.1, A is similar to a positive matrix.

Proposition 4.6 Let A be an n×n irreducible nonnegative matrix with exactly n + 1 positive elements and tr(A) = 0. If A doesn’t satisfy (ii) of Proposition

4.4. Then A is permutationally similar to a irreducible matrix M (1) =                     0 + + 0 . .. 0 ... . .. ... + + . .. 0 0 + . .. + . .. ... . .. + + 0                     , or M (2) =                     0 + + 0 . .. 0 ... . .. ... + . .. + 0 + . .. + . .. ... . .. + + 0                     , or M (3) =                       0 + + 0 . .. 0 ... . .. ... + . .. 0 ... . .. ... + + . .. 0 0 + 0 . .. . .. + + 0                       .

Proof. Since A = [aij]ni,j=1 is irreducible with exactly n + 1 nonzero entries,

hypothesis, columns j1, j2 and rows i1, i2 are only a zero. Without loss of

gen-erality, we may assume that i = 1, j1 = 2, and j2 = 3. Hence if j = 1, then A

is permutationally similar to M (1). If j 6= 1, then A is permutationally similar to M (2) or M (3).

Proposition 4.7 Let A = [aij]ni,j=1 be a nonnegative irreducible matrix with

exactly n + 2 positive elements and tr(A) > 0. If A doesn’t satisfy (ii) of Proposition 4.4 and the diagonal entries have a nonzero element, ⊕. Then A is similar to a positive matrix.

Proof. By Proposition 4.6, we only need to consider the cases M (i) + ⊕Ekk,

where 1 ≤ i ≤ 3, 1 ≤ k ≤ n, and Ekk is defined as in Proposition 3.3.

The case M (1) + ⊕Ekk:

Suppose that ai16= 0.

(1) For k = 1, 2, 3, by Lemma 2.3 (column), M (1) + ⊕Ekk is similar to a

nonnegative irreducible matrix with positive column k. Hence M (1) + ⊕Ekk is

similar to a positive matrix.

(2) For 4 ≤ k ≤ i + 1, if k is odd, by Lemma 2.3 (column), M (1) + ⊕Ekk is

similar to a nonnegative irreducible pattern matrix B(1) with the form

B(1) =                               0 + + . .. 0 0 0 · · · + + 0 0 0 · · · + + . .. 0 + + 0 + + 0 0 + ⊕ 0 + . .. ... ... 0 0 + + 0 0 0 + . .. ... 0 + + 0                               .

B(1)k ≥ B(1)k−1_{. Then B(1) is similar a nonnegative irreducible pattern}

ma-trix B(2) with the form

B(2) =                               0 + + . .. 0 0 0 · · · + + 0 0 0 · · · + + . .. 0 + + 0 + + 0 + + + ⊕ 0 + . .. ... ... 0 0 + + 0 0 0 + . .. ... 0 + + 0                               .

B(2)k ≥ B(2)k+1_{. Then B(2) is similar to a nonnegative irreducible pattern}

matrix B(3) with the form

B(3) =                                 0 + + . .. 0 0 0 · · · + + 0 0 0 · · · + + . .. 0 + + 0 + + 0 + + + ⊕ 0 + + 0 + + . .. ... ... 0 0 + + 0 0 0 + . .. ... 0 + + 0                                 .

B(3)k≥ B(3)k+2_{, and keep on repeating these steps. Then B(3) is similar to a}

nonnegative irreducible pattern matrix B(4) with the form

B(4) =                                 0 + + . .. 0 0 0 · · · + + 0 0 0 · · · + + . .. 0 + + 0 + + 0 + + + ⊕ 0 + + 0 + + .. . ... . .. ... + + 0 0 + + + + 0 0 + + 0 + .. . ... . .. ... + + 0 + + + + 0                                 .

B(4)k ≥ B(4)1_{, and keep on repeating these steps. Then B(4) is similar to a}

nonnegative irreducible pattern matrix B(5) with the form

B(5) =                                 0 + + + + . .. .._. .._. 0 0 0 · · · + + 0 + + 0 0 0 · · · + + 0 + + . .. 0 + + 0 0 + + 0 0 + + + ⊕ 0 + + 0 + + .. . ... . .. ... + + 0 0 + + + + 0 0 + + 0 + .. . ... . .. ... + + 0 + + + + 0                                 .

B(5) has positive column k and than B(5) is similar to a positive matrix. Suppose that k is even. its proof will be omitted, since it is similar to the proof above.

(3) For i + 2 ≤ k ≤ n, by Lemma 2.3 (column), M (1) + ⊕Ekk is similar to a

nonnegative irreducible pattern matrix P with the form

P =                             0 + + 0 0 0 + 0 0 0 0 0 + . .. ... ... 0 0 + + 0 0 0 0 + . .. ... . .. ... 0 + ⊕ + .. . . .. ... + + + + + + + + +                             .

Since Pn≥ Pk−1, then P is similar to a nonnegative irreducible pattern matrix

P0, where P0k−1= Pn+ Pk−1, P0l= Pl for l 6= k − 1. Repeat these steps, it is similar to a nonnegative irreducible pattern matrix with the form

                            0 + + 0 0 0 + 0 0 0 0 0 + . .. ... ... 0 0 + + 0 0 0 0 + . .. ... . .. ... 0 + ⊕ + .. . . .. ... + · · · + + + + + + · · · + + + + +                             .

It has positive row n. Hence M (1) + ⊕Ekk is similar to a positive matrix.

The cases M (2) + ⊕Ekk and M (3) + ⊕Ekk :

Its proof will be omitted, since it is similar to the proof above.

From Propositions 4.1, 4.3, 4.4, and 4.7, we obtain the following theorem. Theorem 4.8 Let A be an n × n nonnegative irreducible matrix with exactly n + 2 positive elements and tr(A) > 0. Then A is similar to a positive matrix.

References

[1] A. Borobia, J. Moro, On nonnegative matrices similar to positive matrices. Linear Algebra Appl., 266 (1997), pp. 365–379.

[2] T. J. Laffey, Raphael Loewy, Helena ˇSmigoc, Nonnegative matrices that are similar to positive matrices. SIAM. J. Matrix Anal. Appl., 31 (2009), pp. 629–649.

[3] T. J. Laffey, Extreme nonnegative matrices, Linear Algebra Appl., 275/276 (1998), pp. 349–357.