1 comments about Matrix Algebra for Applied Economics.

1. This bood can be the introduction level bood for undergraduate students. Authors explain everything very clearly from first chapters. If you already have background about linear algebra this book could be the supplementry material. If you want to focus on the advance econometrics this book would be the good choice for reinforce your matrix ability.

No comments about Linear Functions and Matrix Theory (Textbooks in Mathematical Sciences).

1 comments about Theory of Matrices.

1. This is a self-contained and extremely well-written and clear exposition of matrix theory primarily from the viewpoint of matrix polynomials. The proofs are all clear and easy to follow without sacrificing completeness. I would recommend reading this book before reading Theory of Matrices by Lancaster and Tismenetsky.

4 comments about Basic Matrix Algebra with Algorithms and Applications (Chapman Hall/Crc Mathematics Series).

1. Good, but COULD be better . . I think the book, having few typos, lacks in concrete explanations ( although quite some are there ) and instead, beats a bit with words around the bush, etc . I interpret some sections of chapters in full clarity entirely, yet still get lots vaguely . Very clear objectives and nice examples with great practice problems; I would recommend more geometrical interpretative topics to aid with a visual relationship of some topics . Overall good .



2. This book is horrendous. I can't believe it's still being published. It claims to illuminate the ideas and claims to be effective in approaching the numerous topics. I've read and reread the chapters, but still the content makes no sense. The worst part about it is that I'm a math major, and so I shouldn't be having this much trouble with it.
   
   It's even worse that I've had the actual author as a substitute teacher numerous times at Colorado State University, and he can't even explain the material concisely in person.



3. I was in one of the first classes Dr. Liebler decided to unleash this atrocity on. It was horrible then with 2 or 3 pages of errata, and it isn't much better now. Just imagine how horrible it was trying to learn the material with this as a textbook and with the author giving lectures that were just as foggy and incoherent. If I could give this book a 0 star rating, I would.



4. I'm in the middle of trying to complete Linear Algebra right now, with Liebler's book. I've never met Dr. Liebler, but all I can say is that his book makes no sense. I don't really have much of a teacher who likes to "explain" things, but rather have you figure them out on your own. And that way is fine and everything, but I seriously do not understand one thing this book is trying to convey. The use of notation and the terms are horrendous. I have never ever had such a difficult time understanding a math book (I'm a math major). I've done well in both my Calculus courses, and have been able to teach myself many applications of math, but this Linear Algebra book is TERRIBLE. If you have ANY choice whatsoever, take a course that uses a different book--you'll be very grateful.

1 comments about Matrices, Geometry & Mathematica.

1. Mathematica based tutorial built largely around Singular Value Decomposition analysis. Gives a very solid intuitive understanding of the subject.

No comments about Special Matrices and Their Applications in Numerical Mathematics: Second Edition: Second Edition.

2 comments about Introduction to Matrix Analysis (Classics in Applied Mathematics).

1. I had to use this book as a text for a class on Matrix Theory. It is really hard to learn much about matrices from this book. It is poorly organized and assumes you already know a lot. Two quarters of graduate level linear algebra was not enough. I actually drew on results from a number of different graduate courses (diffential equations, real analysis, etc.) to help me and still found this book difficult to learn from.

   Many of the sections of the book present key material in a very general way. The proofs are often very condensed and sometimes use more advanced results that can only be found with outside research. Many of the problems are very difficult and require advanced techniques not presented in the book.

   Two positives - it has an extensive bibliography and could be a good source of challenging problems for the more advanced student.




2. I have been using this book for many years as a reference and as a source of interesting diversions. Bellman's typically dense style and leaving some proofs to the reader makes for a lot more material than its 387 pages would indicate. The main topics begin with matrix theory in maxima and minima and quickly get to dynamic programming, differential equations, and stability theory. This is the one good reference I have found on Markov matrices (not Markov processes). The last few topics are control processes, invariant imbedding, and Tychonov regularization. Side topics include Vandermonde matrices, Gaussian quadrature, and much, much more.

1 comments about Matrix Algorithms.

1. Please be warned that this book is heavily designed toward matrix theory, rather than the algorithm itself. Therefore, if you are math-averse, you'd better look for other books that offers much lighter theory such as Numerical Recipes. However, if you are not deterred by Greek letters or complicated formulas and eager to learn the theories behind all of the algorithms, this book is for you. Although the author claims that the intended audience is "nonspecialist", I find that this book is most suitable to scientists or grad students, rather than common programmers.
   
   The author explicitly assumes that you know some programming and some linear algebra. Although chapter 1 explains basic matrix theories, you'll need to possess strong basic matrix knowledge -- such as matrix additions, substractions, transpositions, determinants, inverses -- as the author glosses over those on the very first few pages. Still on chapter 1, the author builds up the theories a lot, both matrix theory and linear algebra. He explains many terms such as rank, norms, decompositions, singularity, etc. He also give some proofs on some theorems. These advanced theories will be developed in corresponding later chapters.
   
   Chapter 2 discusses keywords and notations used in the algorithm, such as "for", "if", etc. The pseudocode looks like Pascal. Here, he also give some samples of "easier" algorithms, such as forward substitution and inversion of triangular matrix. He also explains the classical Big O notation briefly and use the sample algorithms as examples on how to compute the Big O notation out. Then, followed by how matrices are stored in the memory and how to optimize the algorithm based on how we store the matrix in the memory. Then, he discuses, quite elaborately, on how to compute rounding error on algorithms and numerical stability of a given algorithm. This last section of chapter 2 is extremely important for those who demands accuracy. This numerical issues will be thoroughly discussed in the later chapters too.
   
   Chapter 3 discusses Gaussian Elimination, LU Decomposition, Cholesky decompositions and their respective variants. The author explains all theory behind them, complete with proofs, discussion on accuracy / rounding error, and Big-O theoretical performance. The algorithms are presented in Pascal-like pseudocode, whose notation described in chapter 2. The author also discuss how certain variants are more beneficial than others (in terms of speed, numerical stability, etc), again, often complete with proofs (or left as an "exercise" for the reader). Theories from chapter 1 are also revisited and expanded as needed.
   
   Chapter 4 discusses QR decomposition and its variants. Complete with background theory, proofs, discussion on accuracy, etc. Just like in chapter 3. It also discusses updating issues and how to adapt QR to linear solutions.
   
   Chapter 5 discusses rank-reducing decompositions. Modifications of previous decompositions to suit this rank-reducing needs, QLP decompositions, and variants of UTV decompositions. Theories, proofs, and discussion of accuracies are all in, as usual.
   
   Speaking of the theory, it's great. It's thorough and the proofs are there even when I think it's not that necessary. By studying this book, one will understand all the theory behind all the decompositions discussed. Numerical issues are discussed very thoroughly, which in itself justifies the price of this book.
   
   However, my main qualms are:
   
   1. Dearth of real examples
   Many examples discussed in the book are way too abstract. Sometimes there are no examples at all. For example: There are no examples in doing Cholesky decomposition. The author only gives an example when the pivoting is necessary (and thus leads to small modification of the algorithm). What about the one that doesn't require any pivoting? Even with the one that requires pivoting, the examples are not given as step-by-step run through. The reader is assumed to know which several steps of the algorithms have already taken place from one matrix to another. In the case of Hessenberg matrix, there are no examples at all. This book is certainly not intended for "nonspecialists".
   
   2. Way too much theory and proofs
   I'm inclined to say that there are way too much theory than it is necessary. I'm not math-averse at all. I was hoping that this book is more practical as the author claims.
   
   3. BLAS acronym
   Maybe this one is a small annoyance. The author uses BLAS acronym a lot. Such as "xeuib" that stands for "X Equals U^(-1) times B". I'm not familiar with BLAS and I have to keep refering to Figure 2.2 to figure one out.
   
   That being said, this is a solid book for explaining theories behind basic decomposition. Treatment on the programming and pragmatic side is rather lacking. Definitely not for the faint-hearted.

3 comments about Matrix Analysis (Graduate Texts in Mathematics).

1. This book is an expansion of the author's lecture notes "Perturbation Bounds for Matrix Eigenvalues" published in 1987. I have used both versions for my students' projects. The book under review centers around the themes on matrix inequalities and perturbation of eigenvalues and eigenspaces. The first half of the book covers the "classical" material of majorisation and matrix inequalities in a very clear and readable manner. The second half is a survey of the modern treatment of perturbation of matrix eigenvalues and eigenspaces. It includes lots of recent research results by the author and others within the last ten years. This book has a large collection of challenging exercises. It is an excellent text for a senior undergraduate or graduate course on matrix analysis.



2. This book is fascinating! Bhatia has made an excellent selection of topics. It is frequently cited in the quantum information literature, and I assume also in the literature of other research subjects. This is on matrix analysis, and it has the flavor of finite-dimensional functional analysis. It is concise and has a very interesting selection of topics.
   
   I have a few suggested tweaks for future(?) editions or classroom discussions:
   
   Remarks on chapter 2:
   
   The presentation at the beginning of chapter 2 would be more motivated if one operationally defines x to majorize y iff y = Ax for some doubly-stochastic matrix A. Bhatia uses an algebraic definition and then proves the equivalence after six pages later. Immediately giving an unmotivated algebraic condition robs the reader of the chance to discover or prove the condition for himself.
   
   There is a very confusing typo in the proof of theorem II.2.8. The statement
   
   "Let r be the smallest of the positive coordinates of x"
   
   should read
   
   "Let r be the smallest of the positive coordinates of y".
   
   Another small remark: Just after the statement of Corollary II.3.4 Bhatia states that "one part of Theorem II.3.1 and Exercise II.3.2 is subsumed by [Corollary II.3.4]." In fact, they are equivalent! That II.3.1 and II.3.2 imply II.3.4 follows immediately from the following
   
   Observation: If f:R->R and g:R->R are convex and f is monotonically-increasing then f composed with g is convex.
   
   Notes on chapter 4:
   
   It would be nice to have the isomorphism between balls and norms presented, perhaps just as an exercise. Then the reader can get a visual mental picture of the various conditions for a norm to be a symmetric gauge function. It might also be nice to move theorem IV.2.1 to the very beginning of that chapter, so that the reader sees the point of section IV.1 immediately.
   
   A small remark is that the proof of Theorem IV.1.8 is made slightly more transparent by the observation that by Theorem IV.1.6 on has
   
   [Phi(x^p)]^(1/p) = Sup Phi(xz),
   
   where the supremum is over z such that (Phi[z^q])^(1/q)=1. (The Sup is attained when x^p = z^q.) Then Theorem IV.1.8 follows immediately from the triangle inequality and subadditivity of suprema:
   
   [Phi(x+y)^p]^1/p = Sup Phi((x+y)z) <= Sup [Phi(xz)+Phi(yz)] <= Sup Phi(xz) + Sup Phi(yz)
   
   Chapter 5:
   
   Chapter 5 covers some of the most interesting and surprising mathematics I have ever seen.
   
   Remarks:
   
   1. All the regularity needed to classify the matrix monotone functions is already present in the case of 2 x 2 matrix monotone functions. Perhaps concretely classifying them would modularize the parts of a complicated proof, allowing some separation between discussion of operator convexity and monotonicity. (Let f:R->R be non-constant. Then f is 2x2 matrix monotone iff f is differentiable with df/dt>0 everywhere and (df/dt)^(-1/2) concave. Furthermore, the first two estimates of Lemma v.4.1 continue to hold for 2x2 matrix monotone functions.)
   
   2. Theorem V.3.3 has somewhat restrictive assumptions: Let f:R->R be extended to a map on self adjoint matrices using the functional calculus. Then all that is needed to differentiate f(A+tH) at t=0, where A and H are self-adjoint and t is a real parameter, is for f to be differentiable on the spectrum of A. (f could be discontinuous except on spec(A), for example.)
   
   3. I would have liked to have the definition the "second divided difference" of f at the points {a,b,c} to be "the highest-degree-coefficient of the at-most quadratic polynomial P that interpolates f on the set {a,b,c}. When a=b then one choses P such that P'(a)=f'(a) as well. When a=b=c then one also takes P''(a)=f''(a)." This is the point of exercise V.3.7, but it makes for easier reading for the definition to be conceptual and let the exercise be to work out the algebraic consequences.
   
   Furthermore, if desired one can actually avoid this calculation and proceed to the proof of Theorem V.3.10. (Just replace f by interpolating polynomials and evaluate everything by by algebra. It has the flavor of Feynman diagrams.)
   
   4. In Hansen and Pedersen "Jensen's operator inequality," Bulletin of the London Mathematial Society," 35 pp. 553-564 (2003); arXiv:math.OA/0204049 (2002), the original authors of the non-commutative jensen inequality state
   
   "With hindsight we must admit that we unfortunately proved and used [a different formulation of the noncommutatitve Jensen's inequality]. However, this necessitated the further conditions that 0 is an element of I and that f(0) < 0, conditions that have haunted the theory since then."
   
   Bhatia's presentation is somewhat out-of-date because it does not include the more up-to-date Jensen's inequality from the more recent work cited above. (Note that the more recent paper occured after the current 1996 edition of Bhatia was published.)
   
   Furthermore, in the same paper, Hansen and Pedersen also introduce a nice version Jensen's trace inequality. It is the same as their sharper form of Jensen's operator inequality, except that both sides have a trace in front and that the operator convex function f is replaced by an arbitrary (scalar) convex function f:R->R. (f acts on matrices using the functional calculus). In particular, the trace inequality is much simpler to prove and more widely applicable although less powerful.
   
   5. It would be nice in future editions(?) to include a reference to Petz and Nielsen's nice little proof of strong subadditivity of the von Neuman entropy.
   
   Chapter 7:
   
   I would have liked to see section 7.1 replaced with the following theorem statement (very similar to what's already in 7.1), and see it proved without chosing an arbitrary basis. (Using an arbitrary basis makes Bhatia's proof of the C-S theorem a bit messy, but a reformulation avoids that.)
   
   Definition: A unitary map U on a Hilbert space is a planer rotation iff
   U restricts to the identity on a subspace P of co-dimension 2, and P is unitarily equivalent to
   
   cos(t) sin(t)
   -sin(t) cos(t)
   
   on P.
   
   Theorem: Let E and F be distinct subspaces of the Hilbert space H, with dim E = dim F. Then there exists a set of planer rotations {R_i} with the properties that
   
   1. The two-dimensional rotation subspaces of the R_i are mutually orthogonal and intersect E and F. (In particular, the R_i commute.)
   
   2. Each R rotates by an angle theta in (0, pi/2].
   
   3. E is rotated onto F by the product of the R_i.
   
   Furthermore, the collection of angles theta_i is uniquely determined by E and F, including multiplicity. If the angles theta_i are distinct and strictly less than pi/2 then the corresponding R_i are also uniquely determined.
   
   Further remark on chapter VII: There is an error on page 223. The author states "we have a bijection psi from H tensor H onto L(H), that is linear in the first variable and conjugate linear in the second variable".
   
   This is impossible, since (lamda v) tensor w = w tensor (lamda w). In particular, any map that is linear in the first variable is necessarily linear in the second variable. The practice of introducing a map from H tensor H to L(H) is a cause of much ugly basis-invariance-breaking in quantum information theory and consequently should be discouraged.



3. Nice book. Many useful facts combined in one volume. Real pleasure to read it.
   The only drawback is sketchy last chapter (almost no proofs due to the lack of space, I believe).

2 comments about A Course in Metric Geometry.

1. The authors present a self-contained treatment of the geometry of length spaces. They begin with the definition of length spaces, and by the final chapter cover almost all of the material in the well known survey paper by Burago, Gromov, and Perel'man. Overall, the book provides an interesting and and accessible introduction to an important class of spaces (length spaces arise as limits of sequences of Riemannian manifolds). I found it exceedingly interesting that one can build analogues of most of the concepts from Riemannian geometry using only the intrinsic metric. There is great emphasis on exposition in this book; Burago makes great effort to motivate definitions and provide interesting examples. The reader is cautioned, however, that numerous typographic errors are to be found (some in fundamental definitions).

   My only complaint is the uneveness of the treatment. The authors spend two chapters developing some of the basic tools of Riemannian geometry in the setting of surfaces in R^3. It seems unlikely to me that a reader who is interested in length spaces would not already have at least passing aquaintence with Riemannian geometry. This, however, is a minor point, and I can recommend the book very highly.




2. I am an undergraduate math student, and I encountered this book as the primary background text for a summer project in geometry. I had just been through a year or so of studying manifolds and basic Riemannian geometry, but it was my first substantial exposure to the length space approach to geometry as well as notions like curvature/Alexandrov spaces, polyhedral spaces, the hyperbolic plane, and quasi-isometries. It does a number of things very well, but it does possess some comparitively minor flaws that are nonetheless worth noting.
   
   Pros: The definitions and examples are very clearly explained and in a manner highly suitable for self-study. The exercises are challenging and they do a great deal to get the reader's hands dirty and force him/her to really process the main ideas. Most of the proofs are very well written, and the authors do a good job of demonstrating the appropriate tools and outlining the best approaches in the proofs and the examples. The authors also do a good job in general of summoning the neccessary background material without dwelling on review too extensively; you should probably have already seen the Hausdorff measure and the fundamental group before you pick up this book, but the book recalls the main concepts and results (with helpful exercises) in case your memory has faded or don't have much experience with such notions. Even the review of Riemannian geometry, which might be too long for some, is appropriate to firmly establish notation and because hyperbolic geometry and Riemannian curvature are so central to the topic of the text.
   
   Cons: Some of the key definitions and proofs (such as the triangle criterion for nonpositive/nonnegative curvature) contain typos, but for the most part they are sufficiently obvious that a critical reader will be able to figure out how to fix them with minimal confusion. While most of the proofs are easy to follow (especially the more technical proofs where lack of clarity would be a serious problem), some of the more visually-oriented proofs are very poorly written and there are precious few diagrams to compensate - see, for example, the example involving the curvature of cones over circles of various lengths. Also, the book could use some more exercises to accompany the more technical constructions. The exercises that it has are very well-chosen for the purpose of forcing the reader to internalize the relavent concepts and techniques, but some basic definition chasing to help process the harder definitions would make the process go smoother.