Customer Reviews

Matrix Algebra From a Statistician's Perspective

5 star:		(9)
4 star:		(3)
3 star:		(1)
2 star:		(1)
1 star:		(0)

 

 

I am not a statistician, but this book has been my major reference on matrix algebra since I got it. The presentation is a bit dense, but I want to point out that the author actually presents the proofs to essentially _all_ theorems in the book. Perhaps this explains the style. As for the content, I find this book very comprehensive in my experience. But the dense page-setting of the book actually makes it visually challenging to locate a result. I also note that there are extensive exercises at the end of every chapter, although I probably won't use this as a textbook for my students.

This book is a true rarity. The exposition is very mathematical, and, therefore, many mathematicians (interested in matrix algebra) will find this book very useful and interesting. The exposition is clear, and quite complete. Of great interest are topics such as idempotent matrices, differentiation of matrices, invertibility.

I am a PhD level graduate student who has never had a matrix algebra course. I got this book to help with a Linear Models course I am currently taking. The book is very helpful, and provides a solid background in the first few chapters before building on them for more complex results. Definitely a good reference to have nearby.

There are tons of books on linear algebra, but very few with the scope of this one. Researchers facing any non-trivial task in linear algebra would do well to look here first. There are, for example, chapters on how vectorizing matrices (by stacking the columns) relates to Kronecker products, and taking derivatives of matrix functions. If you need to take the derivative of the determinant of a matrix with respect to its inverse, look no further. I wish the book was typeset better, but the content is fantastic.

Original review:
I'm currently using this book in a class I'm taking. Overall, the content of the book is very solid, and I can see keeping this book (or possibly the hardcover version) on my shelf for years to come. However, the material is very dense and the exposition is generally lacking. Reading this book is difficult due in part to the poor layout decisions that were made; the layout isn't atrocious, but there is significant room for improvement. Also, the soft-cover seems to not want to stay closed (just a minor annoyance).

If you've never taken a matrix algebra course before, this is not the book to learn from (try either the Hoffman and Kunze or Friedberg books - both are considered good undergraduate-level texts). If you are looking for a book to act as a reference, this is a good choice. In my opinion, there should be a somewhat larger focus on the applications of the matrix algebra to statistics.

(4 stars because it is a solid reference and I knew that is what it aims to be - it lost a star due to the layout and cover issues as well as the dissatisfying lack of direct applications to statistics).

Updated:
As the semester progressed and the material covered in the book moved further from material I knew, I became more and more dissatisfied with it. Learning matrix algebra from this book would be like learning English from a dictionary. There are VERY few examples (asymptotically 0?) and very little explanation of what everything relates to.

Here is an example of exposition leading up to a theorem which I would say characterizes 90% of the book:
"The following theorem, which extends the results of Theorem 14.12.19, is obtained by combining the results of Theorem 14.12.32 with those of Theorem 14.12.26 and Corollary 14.12.27."
That's it. No other commentary, explanation of the purpose of the theorem or why it is important or how it relates to anything else. No theorems are highlighted as being more important than any other.

Moreover, the typesetting in the book is among the worst I've seen in a textbook - it really is very difficult to read more than a single theorem and proof.

As such, I've changed my rating to 2 stars.

It is difficult to find an advanced book on matrix algebra. In any discipline that requires numerical computation, one needs knowledge of advanced matrix algebra. Analytical matrix algebra does not have a natrual home. Almost all linear algebra books are too low of level. There are books on numerical linear algebra such as Trefethen and Bau or Golub and Van Loan and books on vector space linear algebra such as Hoffman and Kunze, but neither of these types of books provide broad coverage of advanced matrix algebra. This book fills that gap. I consider this book to be superior to Applied Matrix Algebra in the Statistical Sciences by Alexander Basilevsky, Matrix Algebra: Theory, Computations, and Applications in Statistics by Gentle, and A Matrix Handbook for Statisticians by Seber.

As one reviewer notes, the book does not have a lot of problems. I would focus more on proving the theorems rather than the number of problems. Harville could have proved less of the theorems, and inserted them as problems, but he proved a large number of theorems in detail.

The book is very dense and detailed, which makes it an excellent reference book. However, I don't believe this book is suitable for learning Matrix Algebra. Each chapter includes only a few practice problems which I found to be a major shortcoming. I still gave it a 4/5 because it will still be very helpful for me as a reference.

To get this out of the way first and foremost, this book simply WILL NOT STAY OPEN for at least the first 150 pages. I realize this isn't a real criticism about the material, but it is annoying enough to mention. I've never seen a hardcover book be this stubborn about staying on a page. I've tried everything from weighing it down with something reasonably heavy to stomping on the spine. As soon as I set it down, it closes.

---End Rant---

I was assigned this book for a matrix algebra course, the idea being to get incoming graduate students ready for linear models by patching up any holes in linear algebra. Towards that end, working through this book seems inefficient. It's supposed to be from a statistician's perspective, yet somehow eigenvalues/eigenvectors and the Spectral Theorem aren't touched until 21 chapters in. I find it a little odd that nullspaces aren't defined until 11 chapters in (most texts would address this by chapter 2 I think) and the closest thing to an application comes in chapter 12 with the discussion of projection matrices. I can't decide whether I like or dislike the fact that the book basically ignores computational aspects (e.g. you won't find anything about putting a matrix in reduced row echelon form in here, and very little discussion on, say, the practical ways to invert a matrix).

A unique aspect of this book, compared with other Linear Algebra texts, is the level of abstraction. Everything is at the level of the vector space R^(m x n), which I suppose allows for the discussion of more specialized topics without having to specify. In my opinion, it's pretty comprehensive at this level of generality and covers many topics that are omitted in more standard texts. As far as the general writing of the book, I feel that a lot of the material is under motivated, which is fine for a reference but not good for an assigned textbook.

I imagine that I will keep this book as a reference, particularly for the less essential material. It's well organized and, for my needs, comprehensive enough.

This book is a must to have in your shelf if you deal more than once with matrices. Do not expect to learn from it as from a textbook but it is great as a reference - easy to find any missing bit of knowledge one might miss. All the properties with necessary proofs are there.
Strongly recommended for everybody doing research in statistical learning.

Linear models and multivariate statistics are fundamental topics in statistics. A good understanding of both requires a command of linear algebra. Although several matrix-algebra texts exist, it is helpful to have one that approaches this subject from a statistical perspective. Harville's text is one of these. In 614 pages, it starts from first principles and progresses to advanced topics. It is self-contained, and James Schott -- author of Matrix Analysis for Statistics -- noted that Harville never describes a topic as 'beyond the scope of the text'. Schott wrote this in favorable book review published in JASA (Sept, 1998). To study this text, some familiarity with linear algebra will be helpful. I recommend this book, but its completeness can also be a challenge. Every chapter builds heavily on results from previous chapters. It would be difficult to understand any single topic in isolation because most explanations refer to several proofs covered in earlier chapters. If you do start from the beginning, the material progresses logically. I struggled with some proofs, but that will happen with any text, and overall the proofs are clear. Reading this text is an investment and there are no quick answers. Over time - it could take several months -- you will understand linear models and other statistical concepts in greater depth. Harville's book differs from other similar texts in that it does not discuss statistics. The focus is strictly on matrix algebra relevant to statistics. There is a companion text that contains solutions to the exercises and additional problems.