2 comments about Matrix Analysis for Scientists and Engineers.

1. This book covers all the useful material that you probably didn't get to in your first linear algebra course -- pseudo inverses, SVD, etc. Moves quickly, but not too quickly. Good balance of theory and application.



2. A good review of the main topics in linear algebra. Succinct and to the point. It is good as a reference, but if you are looking for an introduction, please read Strang's books on the topic.

1 comments about Matrix Theory.

1. This book is a little, wonderful gem. Assuming that you know some basic stuff (that you have taken a linear algebra course and know how to deal with matrices), this will help you a lot to have a feeling of "matrix analysis".(but not linear algebra).
   
   Matrices are widespread in all aspects of science (and especially in computer science) and if your research or work is about processing and analysing large amount of data, you cannot avoid dealing with them. I'm a graduate student in computer scince and my advisor told me that, having read that book carefully I would be equipped with basic necessary skills of matrix theory. (Yep, still reading) By the way, my suggestion is that you may skip the third chapter about differential equations and the last chapter about numerical analysis if you are not interested in these topics. But, chapters 1, 2 and especially 4, 5 and 6 are crucial.
   
   This book is good as self study material for math, computer science, electrical engineering and decision science students who has taken suitable undergrad courses. If you are talented in math though, you may want to see the material even in undergrad or in high school. The book presents the material in a theorem-proof style which is quite nice and solid. And if you want to pursue more advanced matrix theory, you may go for a bigger book like Van Golub's book, after digesting the material in this one.
   
   And, the best thing is that it's small and cheap. Definitely 5 stars.

2 comments about Matrix Algebra Useful for Statistics (Wiley Series in Probability and Statistics).

1. This is an excellent introductory book, although mostly oriented to people with a reasonably good mathematical background. Searle covers most, if not all, the matrix algebra that you may need to fully understand applications of mixed linear models. Chapters one to eight cover the essential operations (addition, multiplication, determinants, etc) as well as more advanced concepts (rank, canonical forms, inverse, etc). The rest of the book (chapters nine to fifteen) covers applications to statistics as well as advanced topics that can help you to understand how do some of the statistical packages work inside. The only mathematics book that I have read from cover to cover. Definitively a classic.



2. This book covers a lot of materials that will contribute to a solid understanding of matrix algebra. I would, by all means, recommend this book, but would still like to point out some weaknesses:
   
   1. Unattractive page layout. Page after page of dull looks. Need more visuals; I don't mean colorful graphics but black-and-white diagrams and geometric interpretations.
   
   2. From time to time, I felt like the author would give examples only on easy-to-understand topics, but avoid giving examples on difficult-to-understand ones. That didn't make sense to me.
   
   3. Proofs should be more clearly presented. It seems like the author is always in a hurry to just get the job done. I guess it was designed for uninterested readers to simply skip the proofs without making them feel they have skipped over a lot of materials. However, for an interested reader like me, who isn't particularly strong in proofs, it was a disfavor.
   
   4. Illustration topics could be more diverse. For instance, there are too many illustrations dealing with genetics. I would appreciate more everyday examples, like the taxi one.
   
   Again, I would strongly recommend this book, despite what I think are its weaknesses.

No comments about Matrix Algebra: An Introduction (Quantitative Applications in the Social Sciences).

3 comments about Iterative Methods for Sparse Linear Systems, Second Edition.

1. This is a great book for this subject. The book is easy to follow and Saad does a wonderful job of illustrating with examples. This is a great textbook or a book for reference. This book does a particularly good job with Krylov methods and does a reasonable job with preconditioning.



2. This is one of my favorite books in my library on this subject. Also I have used this book for my class as main textbook along with "Iterative Methods for Solving Linear and Nonlinear Equations" by C. T. Kelley , which is another SIAM book.
   Highly recommended.



3. We used this book to prove a theorem in our studies that is directly related to my PhD thesis on spatial data mining and spatial statistics. This book is a master-piece.
   Thanks Dr. Saad.

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 Matrix Algebra: Exercises and Solutions.

1. The book consists of a seemingly endless number
   of exercises whose solutions give little or
   no insight into the structure underlying the
   problems. The final result is that you are no
   smarter at the end than at the beginning.



2. This book is a companion text to the author's main text on Matrix Algebra. The only way to master matrix algebra is by working through exercises. Most texts have exercises, but few offer solutions. Harville's main text is great because it offers proofs for most theorems. Others are left as an exercise. If we get stuck, we can check the solutions for these results and many more.

4 comments about Matrices and Linear Transformations: Second Edition.

1. Charles Cullen deserves to pat himself on the back for this one. The first three chapters are the meat of the book which review or, possibly to some, introduce the fundamentals of linear algebra (matrices, vector spaces, and determinants.)

   After firmly laying down a foundation from which to work with, Cullen does a remarkable job explaining linear transformations and eigenvalues and eigenvectors (although, some basic calculus is assumed known).

   After introducing matrix similarity and Jordan-canonical form, Cullen dabbles in polynomial matrices and more similarity with divisor theorems and polynomial matrix canonical form.

   Finally, rounding third plate are an intro into formal matrix analysis (in which some upper-level, although not necessarily advanced, calculus is necessary) and numerical methods. Although these two chapters only encompass about 35 pages total, they do offer a solid foundation for further study.

   I recommend this book as it breaks a lot of steps involving matrix calculations down and allows the reader to not only understand how, but why.




2. So many math books take a relatively simple topic and with the use of horrible notation and confused english, convert it to something esoteric and complicated. This book, however, is anything but that. It involves mathematical rigor, covers a broad range of topics and has answers to selected problems at the back which for a math book, works real well for me. Besides you can't argue with the price.
   This book never leaves my desk, recommended strongly.



3. This book is in its second edition. But it was copyrighted 1972, not 1990. Its original price was $8.95. Perhaps Dover increased its price to $14.95 in the latest printing. But I still think Amazon was a bit misleading there.
   
   In any case, this is an undergraduate level reference book with proofs on Laplace Expansion, Cramer's Rule, Jordan Canonical Form, Cayley-Hamilton Theorem, and other stuff that you see it proved once and never want to see it again. The last section even covers numerical methods on matrices. All in all, it got a good mix of rigor and practicality for a book of its kind.



4. This book is concise and possesses a great deal of information on matrices. I had to buy $100+ book for a upper division linear algebra class and it sucked; it contained in 300 very cluttered and long winded pages what this book accomplishes in a little over a hundred pages with far greater depth. It's a highly affordable book and quite well written, which, I guess, is why they don't use this book to teach linear algebra in a 'modern' classroom.

No comments about Positive Definite Matrices (Princeton Series in Applied Mathematics).

3 comments about Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory.

1. Sometimes it is very difficult to find out a matrix theorem that is used in a technical publication.
   
   I have books by Gantmacher, Horn, Meyer (undergraduate), Kunze and Hoffman and Jan Magnus but if you NEED to quickly find a result, open this book. Thousands of matrix theorems and facts.
   
   Very good reference.



2. 
   I found this to be an extremely useful book. Its internal organization, extensive index and tables of symbols make it very easy to find exactly the result you need.
   
   I strongly recommend this book to people working in applied mathematics or engineering.



3. This book will become a classic... Like other reviewers, I have copies of many of the important works on matrix theory, but, for example- Needed a result on simultaneous diagonalization. Where to find it? In "Introduction to Statistical Pattern Recognition", Fukunaga, or do it from scratch, or... in this text, along with pretty much every other result that I could want. So, I'm buying a copy both for its completeness and organization. It'll be the first stop every time I'm scratching my head over a matrix algebra problem. Thanks to Bernstein for making my life a bit easier.