Arthur Buchheim (1859-1888)

The new second edition of Horn and Johnson’s Matrix Analysis, about which I wrote in a previous post, includes in Problem 2.4.P2 a proof of the Cayley-Hamilton theorem that is valid for matrices with elements from a commutative ring and does not rely on the existence of eigenvalues. The proof is attributed to an 1883 paper by Arthur Buchheim.

A few years ago Arthur Buchheim’s work came up in my own investigations into the history of matrix functions and I discovered that he was a mathematics teacher at Manchester Grammar School, which is located a couple of miles south of the University of Manchester, where I work.

In 1884 Buchheim gave a derivation of Sylvester’s polynomial interpolation formula for matrix functions. The original formula was valid only for matrices with distinct eigenvalues, but in 1886 Buchheim generalized it to handle multiple eigenvalues using Hermite interpolation.


Appropriately, Rinehart, in his 1955 paper The Equivalence of Definitions of a Matric Function, cited Buchheim when he wrote

“there have been proposed in the literature since 1880 eight distinct definitions of a matric function, by Weyr, Sylvester and Buchheim, Giorgi, Cartan, Fantappiè;, Cipolla, Schwerdtfeger and Richter … All of the definitions except those of Weyr and Cipolla are essentially equivalent.”

Buchheim studied at New College, Oxford, under the Savilian Professor of Geometry, Henry Smith, and then at Leipzig under Felix Klein. Then he spent five years at Manchester Grammar School, from which he resigned due to ill-health the year before his death.

In addition to his work on matrix functions and the Cayley-Hamilton theorem, Buchheim published a series of papers promoting Grassmann’s methods. In his A History of Mathematics (1909), Cajori notes that

“Arthur Buchheim of Manchester (1859-1888), showed that Grassmann’s Ausdehnungslehre supplies all the necessary materials for a simple calculus of screws in elliptic space.”

He goes on to say that

“Horace Lamb applied the theory of screws to the question of the steady motion of any solid in a fluid.”

thus bringing in another, much more famous, Manchester mathematician about whom I recently wrote.

Sylvester wrote an obituary in Nature in which he stated “I … know and value highly his contributions to the great subject which engaged the principal part of my own attention during the transition period between my residence in Baltimore and at Oxford”.

The best source of information on Buchheim is an article

Jim Tattersall, Arthur Buchheim: Mathematician of Great Promise, in Proceedings of the Canadian Society for History and Philosophy of Mathematics Thirty-first Annual Meeting, Antonella Cupillari, ed, 18 (2005), 200-208.

which lists lists 24 papers that Buchheim published in his short life of 29 years.

Second Edition (2013) of Matrix Analysis by Horn and Johnson


Horn and Johnson’s 1985 book Matrix Analysis is the standard reference for the subject, along with the companion volume Topics in Matrix Analysis (1991). This second edition, published 28 years after the first, is long-awaited. It’s a major revision: 643 pages up from 561 and with much more on each page thanks to pages that are wider and taller. The number of problems and the number of index entries have both increased, by 60% and a factor 3, respectively, according to the preface. Hints for solutions of the problems are now given in an appendix.

The number of chapters is unchanged and their titles are essentially the same. New material has been added, such as the CS decomposition; existing material has been reorganized, with the singular value decomposition appearing much earlier now; and the roles of block matrices and left eigenvectors have been expanded.

Unlike the first edition, the book has been typeset in LaTeX (in Times Roman) and it’s been beautifully done, except for the too-large solid five-pointed star used in some displays. Moreover, the print quality is superb. Oddly, equations are not punctuated! (The same is true of the first edition, though I must admit I had not noticed.)

The new edition is clearly a must-have for anyone seriously interested in matrix analysis.

Note, however, that this book is not, and cannot be without greatly increasing its size, a comprehensive research monograph. Thus exhaustive references to the literature are not given (as stated in the preface to the original edition). Also, in some cases a story is partly told in the main text and completed in the Problems, or in the Notes and Further Reading. For example, Theorem on page 184 compares the Jordan structure of the nonzero eigenvalues of AB and BA (previously a Problem in the first edition), but the comparison for zero eigenvalues is only mentioned in the Notes and Further Reading seven pages later and is not signposted in the main text.

The 37-page index is extremely comprehensive and covers the Problems as well as the main text. It’s not perfect: Sylvester equation is missing (or rather, is hidden as the subentry Sylvester’s theorem, linear matrix equations).

A final point: the References (bibliography) contains several books that are out of print from the indicated publisher but are available in reprints from other publishers, notably in the SIAM Classics in Applied Mathematics series. They are:

  • Rajendra Bhatia, Perturbation Bounds for Matrix Eigenvalues, SIAM, 2007: hard copy, ebook.
  • Françoise Chatelin, Eigenvalues of Matrices, SIAM, 2012: ebook, hard copy
  • Charles Cullen, Matrices and Linear Transformations, Second edition, Dover, 1990: Google Books.
  • Israel Gohberg, Peter Lancaster & Leiba Rodman, Matrix Polynomials, SIAM, 2009: hard copy, ebook.
  • Israel Gohberg, Peter Lancaster & Leiba Rodman, Indefinite Linear Algebra and Applications, Birkhauser, 2005: ebook.
  • Marvin Marcus & Henryk Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover, 1992: Google Books.
  • Stephen Campbell & Carl Meyer, Generalized Inverses of Linear Transformations, SIAM, 2009: hard copy, ebook.

Talk on Accuracy and Stability at Cardiff

My previous post was about the launch meeting of SIAM Student Chapter at Cardiff, at which I gave the opening talk. My talk was titled Accuracy and Stability of Numerical Algorithms and covered rounding of (floating point) numbers, the interplay between precision and accuracy, higher precision computations, and the effect of tiny relative errors on performance profiles.

Here I describe four examples that I gave where rounding, or the choice of rounding mode, can have interesting or surprising (to some) effects.

  • In 2006 Justin Gatlin was credited with a new world record of 9.76 seconds for the 100m. Almost a week after the race, the time was changed to 9.77 seconds, meaning that he had merely equalled the existing record held by Asafa Powell. The reason for the change was that his recorded time of 9.766 has incorrectly been rounded down to the nearest hundredth of a second instead of up as the IAAF rules require.
  • In 2008 the Mail on Sunday got agitated by the possibility that whether or not the UK inflation target of 3% would be exceeded (and it was exactly 3% at the time) could depend on a change of one thousandth of a percent. They realized that since the inflation rate is published to one decimal place, a rate of 3.049 would round down to 3.0% but 3.050 would round up to 3.1% (since ties are rounded up in UK government calculations) and mean the target had been missed.
  • In 1983 the Vancouver stock exchange found that its index had halved over the year since it had been founded. It turned out that the index had been rounded down after every calculation. When the index was recomputed (presumably with round to nearest, though my reference doesn’t say) it doubled.
  • My telephone and cable provider, Virgin Media, wrote to me in 2007 with news about pricing. They had decreased the cost of my cable and line rental package. They had also changed the way calls charges are calculated by “rounding up to the next minute” instead of “charging to the nearest second” as before. They gave the example that “a call that lasts 4 minutes 50 seconds will be rounded up to 5 minutes”. What they didn’t mention is that a call that lasts 4 minutes 1 second will also be rounded up to 5 minutes!

The talk can be downloaded from my website: Accuracy and Stability of Numerical Algorithms.

First Meeting of Cardiff SIAM Student Chapter

One of SIAM’s newest chapters, its 104th, based at Cardiff University, held its inaugural meeting, the SIAM Chapter Day, on January 21st, 2013.

Angela Mihai

Student Chapters of SIAM (The Society for Industrial and Applied Mathematics) are groups based at universities and colleges with the aim of promoting applied mathematics and computational science to young mathematicians. Chapters organize a wide range of activities, including conferences, guest lectures, visits to industry, and social events. They have been an area of growing activity for SIAM in recent years and there are now 108 student chapters worldwide, including 23 outside the United States.

I attended the Cardiff meeting and was one of the speakers, along with Simon Cox (Aberystwyth), Alain Goriely (Oxford) and Matthew Gilbert (Sheffield). The lecture theatre was close to full, with an audience of 70 or so. A significant portion of the audience, and the poster presenters, was from the School of Engineering, reflecting the strong links that exist between mathematics and engineering at Cardiff.

Poster session

Angela Mihai is the Chapter’s faculty representative, and also the driving force behind the Chapter being formed. In her opening remarks she mentioned that although this is the Chapter’s launch event, Chapter members have already participated in several events organized by other UK Chapters and the UK & Republic of Ireland SIAM Section. All the signs are that this, the first SIAM Chapter in Wales, will be a great success.

Why and How to Set up a SIAM Student Chapter

SIAM provides funding to support activities, awarding over $36,000 to 74 Student Chapters in the 2012-13 academic year, and it offers support for a Chapter representative to attend the SIAM Annual Meeting and meet with the SIAM leadership. Chapters often collaborate in organizing events, such as the SIAM National Student Chapter Conference held in Manchester in 2012, and these provide an excellent opportunity for networking with like-minded students off campus.

If there is not a SIAM chapter at your institution, it is worth considering setting one up. The process is simple, requiring a letter of intent and petition signed by at least 12 regular or student members of SIAM. If you have trouble getting the signatures I will be happy to help (as will other SIAM officers).

For information on how to set up a Student Chapter see the Student Chapter page on the SIAM website. If you have any questions, contact SIAM’s Membership Manager, Susan Whitehouse, at the address on the web page just mentioned.

See the SIAM Student Blog for ideas on organizing chapter events.

Simon Cox

Top 5 Beamer Tips

Here are my top 5 tips for using Beamer, the popular LaTeX package for producing slides.

1. Navigation Symbols

I’ve seen hundreds of talks with slides prepared in Beamer. I’ve yet to see anyone use the navigation symbols that Beamer puts at the bottom right-hand corner of the screen. These should be turned off with

\setbeamertemplate{navigation symbols}{}

2. Extra Slides

If you include extra slides at the end of your talk, you can stop them affecting the page count (usually shown in the footer) as follows:

\begin{frame}{First Extra slide}

The appendixnumberbeamer package might not be included in your LaTeX distribution, in which case you can download it from CTAN.

3. References

You can include references in your slides using the following code, best placed after \appendix:


4. Sections

In a longer talk it can be useful to break the talk into sections and produce an outline slide at the start of each section. This can be done by putting in the preamble the code


and then typing \section[short_title]{long_title} between frames, where short_title is used for the headline (the line at the top of the screen that can contain various types of information, intended to help the audience know where you are in the talk). In this code I have increased the size of the font used for the outline and made it bold. If you don’t want the headline it can be turned off with


5. Columns

A good way to line up graphics side by side, especially if they are of different heights, is with Beamer’s columns environment:

\begin{columns}[c] % Columns centered vertically.
\column{5.5cm}     % Adjust column width to taste.
\includegraphics ...
\includegraphics ...

A Black Background for More Restful PDF viewing

I find that large expanses of white on my screen can be hard on the eyes, so I’ve customized the colours of my main applications to yellow or white on black. For example, this is what my Emacs window looks like:

Emacs screen capture

I also have MATLAB (via Preferences – Colors) and Thunderbird (via various customizations) set to a black background.

I recently realized that I can get a black background in my two PDF viewers.

Acrobat Pro or Acrobat Reader (Windows or Mac)

Set preferences via

Edit - Preferences - Accessibility - Replace Document Colors 
     - Use High Contrast Colors

which gives the option for white, yellow or green on black, or you can choose a custom background and foreground color.

SumatraPDF (Windows)

Call SumatraPDF with the -invert-colors option, for example as

SumatraPDF.exe -reuse-instance -invert-colors paper.pdf

This is an excellent PDF viewer for use with LaTeX, as it refreshes the view when the PDF file changes, unlike Acrobat.

Skim (Mac)

Unfortunately, for Skim it only appears to be possible to change the colour of the border, not the background or text.

Horace Lamb Portrait in Alan Turing Building

A portrait of Sir Horace Lamb (1849-1934), FRS, Beyer Professor of Pure and Applied Mathematics from 1888 to 1920, is on display on the Atrium bridge of the Alan Turing building in the School of Mathematics at the University of Manchester.


This is the School’s common room, where we meet for morning coffee and lunch and which is the focal point of the School.

The 1913 portrait, approximately, 4 feet by 4 feet, is by Lamb’s son, Henry Lamb, a distinguished painter, and was presented to the University by Ernest Rutherford. It’s difficult to photograph due to reflections on the glass, so I took the photo from an angle.

Lamb made important contributions to many topics in applied mathematics, including waves, acoustics, elasticity, fluid dynamics, with applications to areassuch as seismology and the theory of tides. He is perhaps best known for his book Hydrodynamics, first published in 1879 (under the original title “Treatise on the Mathematical Theory of the Motion of Fluids”), which went through six editions. The second edition (1895) has been digitized by Google and can be downloaded from The Internet Archive.

The School’s main meeting room is named the Horace Lamb Room and contains an ornate writing desk and display cabinets presented to Lamb by the University of Adelaide, where he worked for nine year before moving to Manchester. The cabinets contain the engravings pictured below.

H. Lamb

A.D. 1898

The interesting story of how Lamb, born in Stockport near Manchester, came to take a chair in Adelaide, and why he subsequently returned to Manchester, is told in Horace Lamb and the Circumstances of His Appointment at Owens College by Brian Launder (2013).

For more about Lamb see The MacTutor History of Mathematics archive.

Fine-Tuning Spacing in LaTeX Equations

For several years I’ve been fine-tuning the spacing in LaTeX equations using a tip from Donald Knuth (the same tip works with TeX). He mentioned it in a question and answer session that he gave in 1996 in Holland and which is written up in Digital Typography (1999, p. 641). The idea, which he uses in The Art of Computer Programming, is to write

% Make @ behave as per catcode 13 (active).  TeXbook p. 155.

Then within a formula the @ symbol acts as a macro that inserts 1mu of space. A mu is a math unit. For comparison, a thinspace,


represents 3mu and a \quad 8mu.

Knuth gives the example \sqrt{@\log n} to produce the right amount of space before the l of log.

Some examples of my use of @ from my Functions of Matrices book are

\sqrt{A}^{@@\pm 1}f(\sqrt{A})
f(uv^*) = f(0)I + f[v^*u,0] @@ uv^*
p_{km}(x) = \sum_{j=0}^k \frac{ (k+m-j)!@@ k! }{ (k+m)!@@ (k-j)! }

There aren’t really any rules; the use of @ is largely a matter of taste as to what looks right.

This level of fine-tuning is not something I do as a matter of course in drafts, or with any rigour in papers, but for a book I think it is well worth the effort.

Knuth also points out that $n^2/3$ looks better with a negative thin space, as $n^2\!/3$, so that there is less space between the superscript and the slash. On checking the LaTeX source I notice that I did this in my book, but had forgotten about it until I wrote this post!

SIAM Books on Google Play

In 2011 SIAM launched an institutional e-book program, which makes SIAM books available by chapter in PDF form for readers at subscribing institutions. As of late 2012, SIAM books are now available for individual e-book purchase from Google Play, for use on tablets, smartphones, e-readers, or computers (but not Kindles). Unlike in the institutional program, these e-books are subject to full digital rights management (DRM), which means users cannot copy them or print from them and only the Google account holder has access to the book.

I’ve used the Preview facility to look at a few books on Google Play. My own SIAM books, such as Functions of Matrices (2008), are shown as “scanned pages” and appear to have been scanned from the hard copy; zooming in is supported.


By comparison, the Princeton Companion to Mathematics can be viewed as “scanned pages” or “flowing text” (ePub format). In the latter, which reformats as you zoom in and seems to be the default, the mathematics renders poorly; this is a shame given the impeccable LaTeX typesetting of the original book.

Is there a good solution yet for how to render mathematics in e-books?

Trefethen’s Approximation Theory and Approximation Practice

This new 305-page SIAM book by Nick Trefethen presents a modern approach to approximation by polynomials and rational functions. Much of the theory here underlies the Chebfun software package and almost every page of the book contains examples computed using Chebfun.


The book is certainly a must-read for anyone interested in numerical computation. But the most unusual feature of the book is not immediately obvious: it was entirely produced from 29 MATLAB M-files, one for each chapter. Each M-file contains the book’s text in comment lines intertwined with the MATLAB code that generates the examples and the figures. The book was created by using the MATLAB command publish to generate LaTeX output, which was then run through LaTeX (with a few tweaks for the actual printed book). Nick has made the M-files available at the book’s web page and you can generate the book by running them all through publish.

When I ran publish on one of the M-files it gave a strange error beginning

No method 'createTextNode' with matching signature found for class

and I got the same error whatever M-file I tried to publish. This seems to be caused by a clash with some nonstandard M-file on my path, because if I reset the MATLAB path with the matlabrc command (and then add back chebfun to the path) everything works fine.