## 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 A version of the article with an extended bibliography containing additional historical references is available as a MIMS EPrint.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.