1814 – 1897
The Poet of Mathematics: Matrices, Invariant Theory & the American Journal
James Joseph Sylvester was born on 3 September 1814 in London to a Jewish family named Joseph. He adopted "Sylvester" as a surname later in life.
He entered St John's College, Cambridge, in 1831 and placed Second Wrangler in the 1837 Mathematical Tripos — but was denied his degree because, as a Jew, he could not subscribe to the Thirty-Nine Articles of the Church of England.
This religious discrimination would shadow his career for decades. He could not hold a fellowship at Cambridge or teach at Oxford until the religious tests were abolished in 1871.
He received his BA and MA from Trinity College Dublin instead, and began teaching at University College London in 1838.
Despite being Second Wrangler, Sylvester was barred from Cambridge degrees until 1872. This fuelled his restless career across continents.
At age 14, he was already studying under Augustus De Morgan at University of London. His talent was recognised immediately.
Sylvester was also a poet, publishing "The Laws of Verse" (1870). He saw deep connections between mathematical structure and poetic form.
Sylvester's career was remarkably peripatetic. After UCL, he spent an unhappy stint at the University of Virginia (1841–42), leaving after a confrontation with a student.
Back in London, he worked as an actuary and studied law at the Inner Temple, being called to the bar in 1850. During this period he met Cayley, beginning one of mathematics' greatest collaborations.
In 1855, he became professor at the Royal Military Academy, Woolwich, where he remained for 15 years. His most creative period followed at Johns Hopkins University (1876–83), where he founded the American Journal of Mathematics.
He returned to Oxford as Savilian Professor of Geometry in 1883, finally receiving the recognition Britain had long denied him.
Meets Cayley while studying law at Lincoln's Inn. Their partnership transforms algebra.
Appointed first professor of mathematics at Johns Hopkins. Brings European research culture to America.
Founds the American Journal of Mathematics — the first research mathematics journal in the United States.
Returns to England as Savilian Professor at Oxford. Finally receives full recognition at age 69.
Sylvester worked during the Victorian era, when British mathematics was reviving after a century of stagnation relative to the Continent.
Britain had lagged behind France and Germany since Newton's era. Sylvester and Cayley led the revival of British algebra, creating an "English school" of invariant theory.
Until 1871, non-Anglicans were excluded from Oxford and Cambridge positions. Sylvester, as a Jew, faced systematic discrimination that shaped his transatlantic career.
Johns Hopkins (founded 1876) was America's first research university. Sylvester's appointment signalled the birth of American research mathematics.
The 19th century saw the emergence of abstract algebra — groups, rings, fields, matrices. Sylvester's terminological innovations helped crystallise these concepts.
Invariant theory dominated algebra from 1840–1890. Sylvester and Cayley's work competed with that of the German school led by Clebsch and Gordan.
Sylvester coined hundreds of mathematical terms: matrix, discriminant, minor, Jacobian, Hessian, covariant, and many more that remain in daily use.
Invariant theory studies properties of algebraic forms that remain unchanged under linear transformations. Sylvester and Cayley developed this into a systematic discipline.
Given a binary form f(x,y) = a_0 x^n + ... + a_n y^n, an invariant is a polynomial in the coefficients that is preserved (up to a power of the determinant) under the substitution x → αx + βy, y → γx + δy.
Sylvester proved foundational results about the structure of invariant rings and discovered key syzygies — algebraic relations among invariants.
Sylvester's approach to invariant theory was combinatorial and constructive. He developed methods to enumerate invariants and covariants of binary forms of any degree.
His key innovations included the partition method for counting invariants, connecting the theory to his work in combinatorics and partition theory.
He introduced syzygies — algebraic relations satisfied by invariants — and proved that the ring of invariants is finitely generated for binary forms. This was the British counterpart to Gordan's theorem (1868).
Sylvester's "atomic theory of invariants" attempted to build all invariants from basic building blocks, anticipating modern representation theory and the symbolic method in commutative algebra.
Invariant, covariant, contravariant, discriminant, Hessian, Jacobian, resultant, syzygy — all coined or popularised by Sylvester.
Gordan proved finite generation constructively (1868). Hilbert's non-constructive proof (1890) shocked the invariant theory community. Sylvester preferred constructive methods.
Invariant theory was revived by Mumford's geometric invariant theory (1965) and remains central to algebraic geometry, representation theory, and physics.
Sylvester determined the number of linearly independent invariants and covariants for binary forms of low degree — a precursor to modern Hilbert series computations.
Sylvester coined the word "matrix" in 1850, defining it as a rectangular array of numbers from which determinants could be formed. The term was inspired by the Latin word for "womb" — a matrix gives birth to determinants.
He also introduced the terms minor, rank, and nullity for matrices, establishing the basic vocabulary of linear algebra.
Sylvester proved the law of inertia for quadratic forms: the number of positive, negative, and zero eigenvalues is invariant under congruence transformations. This is now called Sylvester's law of inertia.
While Cayley developed the algebra of matrices (multiplication, inverse, etc.), Sylvester focused on the spectral theory and canonical forms of matrices.
Sylvester's law of inertia (1852) was a landmark: for a real symmetric matrix, the numbers of positive, negative, and zero eigenvalues are invariant under congruence. This signature (p, q, z) classifies quadratic forms up to equivalence.
He also proved results on the characteristic equation of a matrix and studied what we now call Sylvester matrices — structured matrices used to compute resultants and test polynomial coprimality.
The Sylvester equation AX + XB = C is fundamental in control theory and numerical linear algebra.
Used to compute the resultant of two polynomials. If the Sylvester matrix is singular, the polynomials share a common root. Essential in elimination theory.
Sylvester contributed to the theory of latent roots (eigenvalues), proving bounds and studying their behaviour under perturbation.
Modern linear algebra textbooks still use Sylvester's terminology: matrix, rank, nullity, minor, signature. His vocabulary became the standard.
Sylvester made important contributions to partition theory and combinatorics. He proved constructive versions of Euler's partition theorems and developed graphical methods for studying partitions.
His work on Farey sequences and graph theory anticipated developments that would not be fully explored for decades. The Sylvester-Gallai theorem states that a finite set of non-collinear points determines at least one line through exactly two points.
In 1878, Sylvester founded the American Journal of Mathematics, the first American research journal in mathematics. He served as its editor-in-chief and used it to publish both European and American mathematical work, transforming the American mathematical landscape.
Developed graphical representations of partitions (Ferrers-Sylvester diagrams) and proved bijective results connecting different partition identities.
Founded 1878 at Johns Hopkins. Still published today by Johns Hopkins University Press. The oldest continuously published math journal in the Western Hemisphere.
Every finite set of points not all on a line has a line passing through exactly two of them. Proved by Gallai (1944) after Sylvester conjectured it (1893).
Coined "graph" in the chemical-graph-theoretic sense. Connected graph theory to invariant theory through "chemistry of forms."
"Mathematics is the music of reason."
— James Joseph SylvesterCoin precise terminology for new concepts
Count invariants, partitions, configurations
Build objects explicitly via combinatorial methods
Seek the broadest algebraic framework
Sylvester's mathematical style was exuberant, verbal, and combinatorial. He thought through language — creating names helped him think about structures. His papers are famously discursive, with long footnotes, poetic digressions, and passionate advocacy for mathematical beauty. Where Cayley was concise, Sylvester was expansive; where others were abstract, Sylvester was constructive.
Sylvester's partnership with Cayley, forged at Lincoln's Inn, produced the most fruitful collaboration in 19th-century algebra. Their "invariant theory" shaped a generation.
Sylvester faced systematic discrimination as a Jew in Victorian England. Despite being Second Wrangler, he was denied his Cambridge degree, all Cambridge and Oxford positions, and many other opportunities.
His brief time at the University of Virginia ended when a student attacked him with a bowie knife after Sylvester reprimanded him. Sylvester defended himself with a sword-cane and left America for decades.
He spent years working as an actuary and lawyer, feeling his mathematical talents were being wasted. Only at age 41 did he secure the Woolwich professorship.
Yet Sylvester's irrepressible spirit and productivity never flagged. He published major papers into his 80s and was still lecturing at Oxford at age 80.
The Universities Tests Act of 1871 finally opened Oxford and Cambridge to non-Anglicans. Sylvester received his Cambridge MA in 1872, 35 years after earning it.
In 1842, a student who Sylvester had disciplined attacked him. The university backed the student. Sylvester left, disillusioned with American academic culture of that era.
Copley Medal (1880), Royal Society fellowship, Savilian Professorship (1883). The honours came late but in abundance.
Matrix, rank, nullity, minor, signature — Sylvester's terminology is the language of linear algebra, used by every mathematician, physicist, and engineer.
Mumford's GIT (1965) revived invariant theory. Sylvester's constructive approach to computing invariants is now central to computational algebra.
Partition theory, Ferrers diagrams, and the Sylvester-Gallai theorem remain active research areas in discrete mathematics.
By founding the AJM and mentoring a generation at Johns Hopkins, Sylvester launched American research mathematics. The journal he created still publishes today.
Sylvester's "atomic theory" of invariants anticipated the decomposition of polynomial representations of GL(n), now central to representation theory.
Sylvester coined more mathematical terms than any other individual. His gift for naming made abstract concepts concrete and communicable.
The Sylvester equation AX + XB = C is fundamental in control theory, stability analysis, and model reduction. Efficient algorithms for solving it underpin modern control systems.
Sylvester matrices are used to compute resultants and GCDs of polynomials, fundamental operations in symbolic computation systems like Mathematica and Maple.
Sylvester's law of inertia classifies quadratic forms in special relativity (the metric signature), quantum mechanics (Hermitian operators), and continuum mechanics.
Sylvester-type Hadamard matrices are used to construct error-correcting codes and signal-processing transforms (Walsh-Hadamard).
Karen Hunger Parshall (1998). The definitive biography, drawing on extensive archival research. Covers his mathematics, personality, and struggles with discrimination.
Karen Hunger Parshall (2006). Explores how Sylvester's Jewish identity shaped his career and mathematical contributions across three countries.
Igor Dolgachev (2003). A modern treatment of classical invariant theory that traces ideas back to Sylvester and Cayley.
Cambridge University Press (4 volumes, 1904–12). Sylvester's complete works, showcasing his extraordinary range and verbose style.
"The object of pure mathematics is not the discovery of truth but the creation of concepts."
— James Joseph SylvesterJames Joseph Sylvester
1814 – 1897
The poet of algebra who named the matrix, championed invariants, and gave America its first research journal.