James Joseph Sylvester (1814–1897) Bicentenary

This year (or more precisely September 3, 2014) is the bicentenary of the birth of James Joseph Sylvester, FRS, a prolific 19th century mathematician who led an eventful life, holding positions at five academic institutions, two of them in the USA.


My article Sylvester’s Influence on Applied Mathematics published in the August 2014 issue of Mathematics Today explains how Sylvester’s work continues to have a strong influence on mathematics. A version of the article with an extended bibliography containing additional historical references is available as a MIMS EPrint.

In the article I discuss how

  • Many mathematical terms coined by Sylvester are still in use today, such as the words “matrix” and “Jacobian”.
  • The Sylvester equation AX + XB = C and the quadratic matrix equation AX^2 + BX + C = 0 that he studied have many modern applications and are the subject of ongoing research.
  • Sylvester’s law of inertia, as taught in undergraduate linear algebra courses, continues to be a useful tool.
  • Sylvester gave the first definition of a function of a matrix, the study of which has in recent years has become a very active area of research.
  • Sylvester’s resultant matrix, which provides information about the common roots of two polynomials, has important applications in computational geometry and symbolic algebra.

Sylvester’s collected works, totalling almost 3000 pages, are freely available online and are well worth perusing: Volume 1, Volume 2, Volume 3, Volume 4.

In a subsequent post I will write about Sylvester’s life.

David Broomhead (1950–2014)

David Broomhead passed away on July 24th, 2014 after a long illness. David was a Professor of Applied Mathematics in the School of Mathematics at the University of Manchester. I got to know him in 2004 when the Victoria University of Manchester merged with UMIST and the two mathematics departments, his at UMIST and mine at VUM, became one.


David was a truly interdisciplinary mathematician and led the CICADA (Centre for Interdisciplinary Computational and Dynamical Analysis) project (2007-2011), a £3M centre funded by the University of Manchester and EPSRC, which explored new mathematical and computational methods for analyzing hybrid systems and asynchronous systems and developed adaptive control methods for these systems. The centre involved academics from the Schools of Mathematics, Computer Science, and Electrical and Electronic Engineering, along with four PhD students and six postdocs, all brought together by David’s inspirational leadership.

One of the legacies of CICADA is the burgeoning activity in Tropical Mathematics, which straddles the pure and applied mathematics groups in Manchester, and whose weekly seminars David managed to attend regularly until shortly before his death. Indeed one of David’s last papers is his Algebraic approach to time borrowing (2013), with Steve Furber and Marianne Johnson, which uses max-plus algebra to study an algorithmic approach to time borrowing in digital hardware.

Among the other things that David pioneered in the School, two stand out for me. First, he ran one of the EPSRC creativity workshop pilots in 2010 under the Creativity@Home banner, for the CICADA project team. The report from that workshop contains a limerick, which I remember David composing and reading out on the first morning:

One who works on Project CICADA

Has to be a conceptual trader

Who needs the theory of Morse

To tap into the Force –

A mathematically driven Darth Vader!

The workshop was influential in guiding the subsequent activities of CICADA and its success encouraged me to organize two further creativity workshops, for the numerical analysis group and for the EPSRC NA-HPC Network.

At the CICADA Creativity Workshop, November 2010.

The second idea that David introduced to the School was the role of a technology translator. He had organized (with David Abrahams) a European Study Group with Industry in Manchester in 2005 and saw first-hand the important role played by technology translators in providing two-way communication between mathematicians and industry. David secured funding from the University’s EPSRC Knowledge Transfer Account and combined this with CICADA funds to create a technology translator post in the School of Mathematics. That role was very successful and the holder (Dr Geoff Evatt) is now a permanent lecturer in the School.

I’ve touched on just a few of David’s many contributions. I am sure other tributes to David will appear, and I will try to keep a record at the end of this post.

Photo credits: Nick Higham (1), Dennis Sherwood (2).

Updates: Reminiscences and Obituaries

Matrix Functions Course at Gene Golub SIAM Summer School 2013

As described in a previous blog post, I gave a course on matrix functions at the Gene Golub SIAM Summer School in Shanghai last July. Summer Schools are regularly held in Shanghai and it has been traditional to produce a booklet with summaries of the courses delivered. The organizers therefore asked the speakers at the Golub Summer School to provide a summary of their courses.

I have written a summary, jointly with my postdoctoral research associate Lijing Lin, who acted as TA for the course. It is available as Matrix Functions: A Short Course (MIMS EPrint 2013.73). You can also access the other materials for the course.

In 2005 I interviewed Gene Golub when he visited Manchester. A transcription of the interview is available as An Interview with Gene Golub (MIMS EPrint 2008.8).

If you didn’t have the chance to meet Gene the interview will give you some insight into his career and the early history of numerical linear algebra. Here is a photo of Gene that I took after the interview.

Gene Golub, July 2005 by Nick Higham.

The sketch below is by John de Pillis, a Professor of Mathematics at the University of California, Riverside. John is a talented sketcher and cartoonist and his 777 Mathematical Conversation Starters is full of cartoons, stories, and quotes. It includes a quote from Gene:

Most problems in scientific computing eventually lead to solving a
matrix equation.

Gene Golub by John de Pillis, 1991.

The Lanczos Tapes

Charcoal sketch of Cornelius Lanczos by John Chirnside of UMIST.

Cornelius Lanczos (1893-1974) gave lectures at UMIST (a predecessor institution of The University of Manchester) in the late 1960s and early 1970s, while he was a Professor at the Dublin Institute for Advanced Study. In 1972, UMIST Audio Visual Services made three video recordings lasting almost three hours of Lanczos talking about mathematics, his life, and Einstein. In two of the tapes he is speaking in a group discussion, while in the other he speaks eloquently about his life for 50 minutes, directly to camera and apparently without notes. The topics he covers include his experiences as

  • student of Eötvös and Fejér in Hungary,
  • theoretical physicist,
  • assistant of Albert Einstein in Germany,
  • numerical analyst and inventor of the tau method,
  • (re-)discoverer of the fast Fourier transform and singular value decomposition,
  • inventor of the Lanczos algorithm while working at the US National Bureau of Standards, and
  • head of the Theoretical Physics Department at the Dublin Institute for Advanced Study.

The charcoal sketch above hung for many years in the office of the administrator of the mathematics department at UMIST and now has pride of place on the wall in my office in the Alan Turing Building.

My colleague Stefan Güttel has produced a version of the videos with bookmarks linking to the main topics of discussion. We are pleased to make the videos available online on the occasion of the 120th anniversary of Cornelius Lanczos’s birth (February 2, 1893).

Cornelius Lanczos

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.

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.