Cjr Aljabar.docx

Critical Journal Review

Compiled by Meidy Adelina (416 331 2017 ) Paulus Nainggolan (416 331 2021)

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

TABLE OF CONTENT PREFACE TABLE OF CONTENT CHAPTER I PRELIMINARY 1.1 Background 1.2 Purpose IDENTITY OF JOURNAL CHAPTER III 2.1 Summary Of Journal 2.2 Advantages And Weakness Of Book CHAPTER III : CLOSING 

CONCLUSION



SUGESSTION

BIBLIOGRAPHY

CHAPTER I PRELIMINARY 1.1 Bacground The ones that become the ones that mak mak mak mak mak add more add. Critical Journal Reviews for students is a task that must be done where CJR task is a task that has been given by Lecturers to students in every semester. Where the CJR duties have been newly designed based on the recent KKNI at Medan State University, CJR will be an important part of every semester of lectures. The background of the authors chose this title is, seen from the title that is closely related to the sub-subject of differentiation of the material that occurred in the dictate Subject LINEAR ELEMENTER Elementary. Where in this journal has the title of bringing the Negative Compensation in the nonnegative inverse eigenvalue problem and The symmetric nonnegative eigenvalue problem for 5x5 matrices. In an engineering textbook there is a subject on Eigenvalue and Eigenvectors. This is the background of the author chose the journal 2.1 Purpose 1. To know the role of journal. 2. To know the method used in the journal. 3. To find out the results of the research journal. 4. To know the assessment of the journal

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 RESULTS OF RIVIEW JOURNAL A. 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. We

show that certain matrices cannot be principal submatrices of symmetric, nonnegative matrices with spectrum in U1. In Section 3 we give a sufficient condition for a realizable spectrum, for arbitrary n, to be realizable by a positive symmetric matrix. This condition seems to be of independent interest. In Section 4 we analyze the (+, 0) patterns for which there exists an extreme matrix with spectrum in U1. This analysis yields nine possible patterns, which are further discussed in Section 5. It is shown there that seven of those nine patterns do not allow extreme matrices with spectrum in U1. This analysis is concluded in Section 6 with Theorem 4, which describes the 2 patterns that possibly allow extreme matrices with spectrum in U1. We show that one of these patterns does indeed yield realizable points in U1 which have not been known previously. In addition, the discussion of Sections 4 and 5 yields a result about the sparsity of matrices (not necessarily extreme) with spectrum in U1. This sparsity result is the analogous result, for n = 5, of the sparsity result obtained by Laffey for general nonnegative matrices.

In this section we bring some results that will be used to obtain our main results. The first result, due to Laffey, deals with an extreme matrix.

Theorem 2. [9] Let A = (aij ) be an extreme, n × n symmetric, nonnegative matrix. Then there exists an n × n symmetric, nonnegative, nonzero matrix Y = (yij ) such that

(i)

AY = Y A,

(ii)

aij yij = 0 for all i, j.

Consider next the polytope U whose vertices are the points c, d, e, i, l. It is straightforward to check that U is actually a simplex and therefore has 5 maximal faces (obtained by deleting one of the vertices and taking the convex hull of the rest). In particular, let F1 = convex hull of c, d, e, l, (4) F2 = convex hull of d, e, i, l. Note that F1 is contained in S5, and thus we know any point in it is realizable. On the other hand, every σ ∈ F2 satisfies trace(σ) = 5

λi = 0. Thus, if σ is

i=1

realizable the corresponding matrix A must have trace 0 and σ is clearly an extreme spectrum. It seems that the set of 5 × 5 symmetric, nonnegative matrices with trace 0 has to be dealt separately (certainly Theorem 2 is of no help here because we can chose Y = I5). It should also be pointed out that the corresponding problem for 5 × 5 nonnegative matrices has been solved by Laffey and Meehan [10]. Thus we are led to consider here the set U1 defined by U1 = U/(F1 ∪ F2).

B. ADVANTAGES AND WEAKNESS OF JOURNAL  The Advantages of the journal is: 1. Abstarct Clear, so that by reading the abstract alone readers know the results of research. 2. In this journal, its subjectdiscussion complete and detailed. 3. Part of discussion is complete and detailed.  The Weakness of the journal : 1. The language used in this journal still beat around this bush. 2. Still there are words that are less efficient in its use.

CHAPTER III CLOSING

3.1 Conclusion

3.2 Sugesstion Based on the results of the Critical Journal Review that has been done then it can be submitted some suggestions that can be submitted to students and academics who want to be the next researcher: a) For Students Students are expected to play an active role in conducting a research and development as well as making this journal review as a reference to determine the source of knowledge and other scientific approaches that will be used. b) For Other Researchers Review of this journal is still far from perfection, then it should be reviewed further so that it can complete the deficiencies contained in this journal review.