# Five Examples of Proofreading

Every writer has also to be a proofreader, whether it be of his or her own drafts or of proofs sent by a publisher. In this post I will give some real-life examples of corrections to proofs. The problems to be corrected are not all errors: some are subtle aspects of the typesetting that need improvement. These examples should give you some ideas on what to look out for the next time you have a set of proofs to inspect.

## Example 1

The first example is from proofs of one of my recent papers:

The article had been submitted as LaTeX source and it was reasonable to assume that the only differences between the proofs and what we submitted would be in places where a copy editor had imposed the journal style or had spotted a grammatical error. Fortunately, I know from experience not to make that assumption. These two sentences contain two errors introduced during copy editing: the term “Anderson acceleration” has been deleted after “To apply”, and “We denote by unvec” has been changed to “We denote by vec” (making the sentence nonsensical). The moral is never to assume that egregious errors have not been introduced: check everything in journal proofs.

In a similar vein, consider this extract from another set of proofs:

There is nothing wrong with the words or equations. The problem is that an unwanted paragraph break has been inserted after equation (2.6), and indeed also before “Only”. This set of proofs contained numerous unwanted added new paragraphs.

## Example 2

Here is an extract from the proofs of my recent SIAM Review paper (with Natasa Strabic and Vedran Sego) Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block:

We noticed that the word “how” appears at the end of a line four times within seven lines—an unfortunate coincidence. We suggested that the production editor insert a hard space in the LaTeX source between one or more of the hows and the following word in order to force different line breaks. Here is the result as published:

## Example 3

What’s wrong with this example, from a paper in the 1980s?

The phrase “best unknown” should be “best known”!

## Example 4

The next example is from a book:

At first sight there is nothing wrong. But the $9z$ is suspicious: why $9$, and why is this term that depends only on $z$ inside the integral? It turns out that the equation should read

$k(z) \equiv \frac{2}{z} \int_0^1 \tanh\bigl( z \sin(2\pi t) \bigr) \sin(2\pi t) \,dt.$

When you realize that the left parenthesis and the digit $9$ share the same key on the keyboard you can start to see how the error might have been made at the typing stage.

## Example 5

The final example (from a 2013 issue of Private Eye) is completely different and illustrates a rare phenomenon:

If you cannot see anything wrong after a minute or so, click here. This phenomenon, whereby white spaces in successive lines join up to make a snake, is known as rivers of white. The fix, as in Example 2, is to force different line breaks.

# SIAM Annual Meeting 2017 Preview

It’s a month to the 2017 SIAM Annual Meeting at the David Lawrence Convention Center in Pittsburgh. We’re returning to the location of the 2010 meeting. The meeting is co-chaired by Des Higham (University of Strathclyde) and Jennifer Mueller (Colorado State University).

Here are a few highlights and things it’s useful to know. If you haven’t already made plans to attend it’s not too late to register. Be sure to take in the view from the roof of the convention center, as shown here.

## Block Lecture by Emily Shuckburgh

The I. E. Block Community Lecture on Wednesday evening will be given by Emily Shuckburgh on From Flatland to Our Land: A Mathematician’s Journey through Our Changing Planet. Emily, from the British Antarctic Survey, is a co-author of the recent book Climate Change, which she wrote with HRH Prince Charles and Tony Juniper.

## Prize Lectures

As always, a number of prize lectures will be given at the meeting. These include the four-yearly James H. Wilkinson Prize in Numerical Analysis and Scientific Computing, which will be awarded to Lek-Heng Lim. His lecture is titled Tensors in Computational Mathematics. See this article about Lek-Heng.

## Joint with Activity Group Conferences and Workshops

The meeting is held jointly with the SIAM Conference on Industrial and Applied Geometry (GD17) and the SIAM Conference on Control and Its Applications (CT17), in the same location. One registration fee gains you access to all three meetings!

In addition, the SIAM Workshop on Parameter Space Dimension Reduction (DR17) and the SIAM Workshop on Network Science (NS17) are taking place just before and just after the conference, respectively.

## Funding

Funding of mathematics, and other subjects, is in a state of uncertainty under the current US administration. In the minisymposium How Changing Implementations of National Priorities Might Affect Mathematical Funding a panel of representatives from funding agencies will describe the current situation and future opportunities. This is a great chance to hear the latest news from Washington from those in the know.

## Students

SIAM provides a host of activities for students, beginning with an orientation session on Sunday evening and including a career fair, a session on career opportunities in business, industry and government (BIG), and the chance to meet and talk to invited speakers and co-chairs.

## Hidden Figures

An evening session will include Christine Darden, who was one of the human computers included in the book “Hidden Figures” by Margot Lee Shetterly, on which the recent Hollywood movie of the same title was based.

## SIAM Business Meeting

The Business Meeting (Tuesday at 6.15pm) provides an opportunity to hear the president (that’s me!) and SIAM staff report on SIAM’s activities over the past year and to ask questions. The 2017 SIAM Fellows will be recognized, and a reception in their honor follows the Business meeting.

## Website

SIAM is developing a new website. A preliminary version will be available on laptops in the exhibit hall for participants to try. Feedback will be much appreciated and SIAM staff will be on hand to receive your comments.

## Baseball Match

If you are staying in Pittsburgh on the Friday night, consider attending a baseball match. The Pittsburgh Pirates play the St Louis Cardinals at home at PNC Park on Friday July 14. I went to the Friday match after SIAM AN10 and really enjoyed it; the views from the ground are spectacular.

## Twitter

If you are not able to attend you can get a feel for what’s going on by following the hashtag #SIAMAN17 on Twitter.

## Pittsburgh

There’s plenty to do and see in Pittsburgh, as the following images illustrate. As well as the impressive bridges over the Allegheny and Monongahela rivers, and some interesting downtown architecture and murals, there’s the Andy Warhol Museum (a short walk from the convention center). Here are some images I took in 2010.

# A Second Course in Linear Algebra, by Garcia and Horn (2017)

The publication of a new linear algebra textbook is not normally a cause for excitement. However, Roger Horn is co-author of two of the most highly regarded and widely used books on matrix analysis: Matrix Analysis (2nd edition, 2013) and Topics in Matrix Analysis (1991), both co-authored with Charles Johnson. It is therefore to be expected that this new book by Garcia and Horn will offer something special.

Chapter 0 (Preliminaries) summarizes basic concepts and definitions, often stating results without proof (for example, properties of determinants). Chapters 1 (Vector Spaces) and 2 (Bases and Similarity) are described as reviews, but give results with proofs and examples. The second course proper starts with Chapter 3 (Block Matrices). As the chapter title suggests, the book makes systematic use of block matrices to simplify the treatment, and it is very much based on matrices rather than linear transformations.

Two things stand out about this book. First, it lies part-way between a traditional linear algebra text and texts with a numerical linear algebra focus. Thus it includes Householder matrices (but not Givens matrices), QR factorization, and Cholesky factorization. The construction given for QR factorization is essentially the Householder QR factorization, but the existence proof for Cholesky goes via the QR factor of the Hermitian positive definite square root, rather than by constructing the Cholesky factor explicitly via the usual recurrences. The existence of square roots of Hermitian positive definite matrices is proved via the spectral decomposition. It is possible to prove the existence of square roots without using the spectral theorem, and it would have been nice to mention this, at least in an exercise.

The second impressive aspect of the book is the wide, and often quite advanced, range of topics covered, which includes polar decomposition, interlacing results for the eigenvalues of Hermitian matrices, and circulant matrices. Not covered are, for example, Perron–Frobenius theory, the power method, and functions of nonsymmetric matrices (though various special cases are covered, such as the square root of Jordan block, often in the problems). New to me are the QS decomposition of a unitary matrix, Shoda’s theorem on commutators, and the Fuglede–Putnam theorem on normal matrices.

The 16-page index occupies 3.7 percent of the book, which, according to the length criteria discussed in my article A Call for Better Indexes, is unusually thorough. However, there is some over-indexing. For example, the entry permutation consists of 7 subentries all referring to page 10, but “permutation, 10” would have sufficed. An index entry “Cecil Sagehen” puzzled me. It has two page locators: one on which that term does not appear and one for a problem beginning “Cecil Sagehen is either happy or sad”. A little investigation revealed that “Cecil the Sagehen” is the mascot of Pomona College, which is the home institution of the first author.

There is a large collection of problems that go well beyond simple illustration and computation, and it is good to see that the problems are indexed.

Here are some other observations.

• The singular value decomposition (SVD) is proved via the eigensystem of $A^*A$. Personally, I prefer the more elegant, if less intuitively obvious, proof in Golub and Van Loan’s Matrix Computations.
• The treatment of Gershgorin’s theorem occupies six pages, but it omits the practically important result that if $k$ discs form a connected region that is isolated from the other discs then that region contains precisely $k$ eigenvalues.
• The Cayley-Hamilton theorem is proved by using the Schur form. I would do it either via the minimal polynomial or the Jordan form, but these concepts are introduced only in later chapters.
• Correlation matrices are mentioned in the preface, but do not appear in the book. They can make excellent examples.
• The real Schur decomposition is not included, but rather just the special case for a real matrix having only real eigenvalues.
• Matrix norms are not treated. The Frobenius norm is defined as an inner product norm and, unusually, the 2-norm is defined as the largest singular value of a matrix. There are no index entries for “matrix norm”, “norm, matrix”, “vector norm”, or “norm, vector”.
• The pseudoinverse of a matrix is defined via the SVD. The Moore-Penrose conditions are not explicitly mentioned.
• Three pages at the front summarize the notation and point to where terms are defined. Ironically, the oft-used notation $M_n$ for an $n \times n$ matrix, is not included.
• Applications are mentioned only in passing. However, this does keep the book down to a relatively slim 426 pages.

Just as for numerical analysis texts, there will probably never exist a perfect linear algebra text.

The book is very well written and typeset. With its original presentation and choice of content it must be a strong contender for use on any second (or third) course on linear algebra. It can also serve as a reference on matrix theory: look here first and turn to Horn and Johnson if you don’t find what you want. Indeed a surprising amount of material from Horn and Johnson’s books is actually covered, albeit usually in less general form.