Cjr Aljabar Linear.docx

Critical Journal Review

Compiled by Meidy Adelina (416 33 12017) Paulus Nainggolan (416 33 12021)

MATHEMATICS DEPARTEMENT FACULTY OF MATHEMATIC AND SCIENCES STATE UNIVERSITY OF MEDAN 2018

FOREWARD Thanks to God Almighty for the blessing I am able to complete this Critical Journal, and thanks also to the person who made the journal that I have criticized. Finally, like a people who has many limitations, if there’s a mistake the author expect critisim and constructive suggestion that can then be better. My hope and goals in completing this task are useful and can add knowledge to those who read them.

Medan, 16th May 2018

Author

TABLE OF CONTENT FOREWARD TABLE OF CONTENT CHAPTER I : IDENTITY OF JOURNAL 

First Journal



Second Journal

CHAPTER II : SUMMARY OF JOURNAL 

First Journal



Second Journal

CHAPTER III : ADVANTAGES AND DISADVANTAGES 

Advantages



Disadvantages

CHAPTER VI : CLOSING 

Conclution



Solution

BIBLIOGRAPHY

CHAPTER I : IDENTITY OF JOURNAL First Journal Name

: Negative Compensation in the nonnegative inverse eigenvalue problem

Creator

: Alberto Borobia, Julio Moro, Ricardo Soto

Year

: 2003

ISBN

: 0024-3795

doi

: 10.1016/j.laa.2003.10.023

Second Journal Name

: The symmetric nonnegative eigenvalue problem for 5x5 matrices

Creator

: R. Loeewy, J.J. Mc.Donald

Year

: 2003

ISBN

: 0024-3795

doi

: 10.1016/j.laa.2003.10.018

CHAPTER II : SUMMARY OF JOURNAL First Journal A set ∆ of complex numbers is said to be realizable if ∆ is the spectrum of an entrywise nonnegative matrix. It is clear that if a set ∆ of complex numbers can be partitioned as ٨ = ٨1 ∪ · · · ∪ ٨s in such a way that each ٨i is realizable: if Ai is a nonnegative matrix with spectrum ٨i for i = 1, 2, . . . , s then the block diagonal matrix A = + Ai is nonnegative and has spectrum ∆. The purpose of the present paper is to give conditions under which _ is realizable even if some of the ٨i are not realizable, provided there are other subsets ٨j which are realizable and, in a certain way, compensate the nonrealizability of the former ones. To do this, our main tool will be a result, due to Brauer [7] (Theorem 3.1), which shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of real matrices with constant row sums, are the basic ingredients of our technique. This approach was first adopted by Soto [35] in connection with the nonnegative inverse eigenvalue problem (hereafter NIEP), i.e. the problem of characterizing all possible spectra of (entrywise) nonnegative matrices. We begin by introducing the basic concepts and notation used throughout the paper in Section 2. After briefly recalling both Brauer’s theorem and Suleimanova’s sufficient conditions [37], Section 3 contains the proof of a complex analogue of Suleimanova’s result with the negative real semi-axis replaced by the sector {𝑧 ∈ 𝐶: 𝑅𝑒 𝑧 ≤ 0, |𝑅𝑒 𝑧| ≥ |𝐼𝑚 𝑧|} of the complex plane. Section 5 presents our main result, Theorem 5.1, a realizability criterion for sets of complex numbers which can be partitioned in such a way that the negativity of the nonrealizable pieces can be compensated by the positivity of the realizable ones. We must stress that, although the proof is constructive to a certain extent, it does not allow in general, as in [35], to explicitly construct a nonnegative matrix with the given spectrum, Theorem 5.1 may be sometimes hard to use in practice.

A. Preliminary and Notation A set A = {λ0, λ1 , . . . , λn} of complex numbers is said to be realizable if there exists an entry wise nonnegative n + 1 by n + 1 matrix with spectrum A. The set of all realizable sets is denoted by R. If a complex set A = {λ0, λ1 , . . . , λn} is realizable, then the non-real elements

of

A come in conjugate pairs. Hence, the conjugate set A¯ = {λ¯ 0 , λ¯ 1, . . . , λ¯ n } coin-cides with A. Moreover, the Perron–Frobenius theorem (see [26]) implies that if A = {λ0, λ1 , . . . , λn} is realizable then one of its elements, say λ0, is real and such

that λ0 ≥ |λi | for i = 1. . . n.

Therefore, the set A ≡{ A = {λ0; λ1 , . . . , λn } ⊂ C : A = A, λ0 ∈ R, λ0 ≥ λi for any λi ∈ R } Lemma 2.1 (Johnson [15]). Any realizable set is realized in particular by a nonneg- ative matrix with constant row sums equal to its Perron root. Also, we will use that any matrix in CSα has eigenvector e = (1,..., 1)T cor- responding to the eigenvalue α. For simplicity, we denote in what follows by e any vector of the appropriate dimension with all its entries equal to one. Likewise, we denote by e1 = (1, 0, . . . , 0)T the first column of the identity matrix of any appro-priate dimension. B. Brauer’s theorem and a complex Suleimanova-type theorem

As pointed out in the introduction, our main motivation is to exploit the advanta- ges provided by the following result, due to Brauer [7] in the study of the NIEP. Theorem 3.1 (Brauer [7]). Let A be an n × n arbitrary matrix with eigenvalues λ1 , . . . , λn. Let v = (v 1 ,..., vn)T be an eigenvector of A associated with the eigen-

value λk and let q

be any n-dimensional vector. Then the matrix A + vqT has eigen- values λ1 , . . . , λk−1, λk + vTq, λ k+1 ,..., λn. An immediate consequence of Brauer’s theorem is the following useful and well known result: Lemma 3.1. If A = {λ0; λ1 , . . . , λn } ∈ AR and α > 0, then Aα = {λ0 + α; λ1 , . . . , λn } ∈ AR.

Second Journal The nonnegative inverse eigenvalue problem (NIEP) asks when a list σ = (λ1, λ2 , . . . , λn) of complex numbers is the spectrum of an n × n nonnegative matrix. When λ1, λ2 , . . . , λn are real, the symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when is σ the spectrum of an n × n symmetric nonnegative matrix. Both problems are of great interest and many papers have been written about them, only some of which will be mentioned here. Both are not solved for any n such that n ≥ 5. See [5,11] for some necessary conditions for NIEP. Consider now σ = (3, 3, −2, −2, −2). It is not realizable as the spectrum of a 5 × 5 nonnegative matrix by the Perron–Frobenius theory. Consider now σt = (3 + t, 3, −2, −2, −2) for t > 0. Hartwig and Loewy [4] showed that the smallest t such that σt satisfies SNIEP is t = 1. On the other hand it is shown in the Ph.D. thesis of Meehan [13] that there exists 0 < t < 1 such that σt satisfies NIEP. Thus, NIEP and SNIEP are different already for n = 5. Since we consider here only real n-tuples, we can assume that σ = (λ1, λ2 , . . . , λn), where λ1 ≥ λ2≥ · · · ≥ λn. We say that σ is realizable if it is the spectrum of an

n×n

symmetric nonnegative matrix. If σ is realizable then clearly λ1 is the spectral radius, and we may assume without loss of generality that λ1 = 1. It then follows that λn ≥ −1. We define Rn = {σ = (λ1 = 1, λ2 , . . . , λn) : σ is realizable}. Definition 1. The Soules set Sn consists of all σ = (λ1, λ2 , . . . , λn) in Rn which satisfy: There exists an n × n symmetric, nonnegative matrix A and a Soules matrix R such that Rt AR = diag(λ1 = 1, λ2 , . . . , λn). Let µ = (µ1, µ2 , . . . , µn) with µ1 ≥ µ2 ≥ · · · ≥ µn (so we don’t assume here µ1 = 1). Let e be the 1 × n vector of ones. For any x ≥ 0, consider µ − xe. Suppose that µ is realizable. It follows from [9] that there exists a unique d ≥0 such that µ − de is realizable but µ − xe is not realizable for x > d. We say that µ is an extreme spectrum if d = 0. In that case, if µ is the spectrum of A = At ≥ 0, we say that A is an extreme matrix. It is clear that in order to determine Rn it is enough to find the extreme spectra in Rn. Thus, we are led to consider the extreme spectra which lie in U. This subset is obtained from U by removing a union of 2 faces of U, one being part of the Soules set and the other consisting of the trace 0 matrices.

Realizable spectra for Positive Sysmmetric Matrices In this section we state a sufficient condition for a list of n real numbers to be the spectrum of an n × n positive symmetric matrix. Lemma 2. Let A be an n × n nonnegative, irreducible, symmetric matrix with ei- genvalues λ1 > λ2 ≥ · · · ≥ λn. Given any t > 0, there exists an n × n positive, sym- metric matrix with eigenvalues λ1 + t > λ2 , . . . , λn. Proof. There exists a positive, unit vector x such that Ax = λ1x. It is clear now that A + txxt is positive and satisfies the requirements. Theorem 3. Let A be an n × n nonnegative, symmetric matrix with eigenvalues λ1 > λ2 ≥ · · · ≥ λn. Given any t > 0, there exists an n × n positive, symmetric

matrix with

eigenvalues λ1 + t, λ2 , . . . , λn. Proof. If A is irreducible the proof follows from Lemma 2. So we can assume that A is reducible and has in fact the following form: A = A1 ⊕ A2 ⊕ · · · ⊕ Ak,

k ≥ 2,

where each Aj is irreducible or the 1 × 1 zero matrix. Moreover, we can assume λ1 is the spectral radius of A1, and for 2 ≤ j ≤ k, λlj is the spectral radius of Aj for some 2 ≤ lj ≤ n. Choose E > 0 such that E < min{λ1 − λ2 , t }. Corollary 1. Let A be an n × n nonnegative, symmetric matrix with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. Given any t > 0, there exists an n × n positive, symmetric

matrix with

eigenvalues λ1 + t, λ2 , . . . , λn. Corollary 1 improves Theorem 4.1, part 2, of [15].

Corollary 2. Any point in the relative interior of set of realizable spectra, Rn (viewed as a subset of ffin), is realizable by a positive, symmetric matrix

CHAPTER III: ADVANTAGES AND DISADVANTAGES Advantages 

From journal 1, we can conclude that ∪ is the spectrum of a nonnegative matrix. Additionally, we prove a complex generalization of Suleimanova’s theorem.



From journal 2 We get solution to The symmetric nonnegative inverse eigenvalue problem (SNIEP ) for n = 5. We also give a sufficient condition for a list σ to be realized as the spectrum of a symmetric positive matrix.



Contains images and graphs representing some formulas, as well as the second journal



In both journals, they contain concrete formulas and examples



Theory published in the journal first and second journal some understandable

Disadvantages 

In the first journal, there is some incomprehensible vocabulary.



Writing in the first and second journals there are some that make the reader do not understand, for example taking fonts on some formulas

CHAPTER VI: CLOSING Conclution 

A set ∆ of complex numbers is said to be realizable if ∆ is the spectrum of an entrywise nonnegative matrix.



the block diagonal matrix A = + Ai is nonnegative and has spectrum ∆.



the Perron–Frobenius theorem (see [26]) implies that if A = {λ0, λ1 , . . . , λn} is realizable then one of its elements, say λ0, is real and such



that λ0 ≥ |λi | for i = 1

Since we consider here only real n-tuples, we can assume that σ = (λ1, λ2 , . . . , λn), where λ1 ≥ λ2≥ · · · ≥ λn. We say that σ is realizable if it is the spectrum of an n × n symmetric nonnegative matrix.

Suggestion Of the two journals that we criticize, we can conclude that the second journal can be used in the taking and application in eigenvalue and eigenvector material.

BIBLIOGRAPHY Borobia, Alberto. 2003. Negativity compensation in the nonnegative inverse eigenvalue problem. Chile: Elsevier Inc.

Loewy. R. 2003. The symmetric nonnegative inverse eigenvalue problem for 5x5 matrices. Chile: Elsevier Inc.