Arthur Cayley at a glance:

  • Born 1821 in Richmond, Surrey
  • Graduated from Trinity College, Cambridge, in 1842
  • Smith's Prize recipient, 1842
  • Studied both law and mathematics, with mathematics taking priority
  • Accepted mathematics position at Cambridge in 1863, abandoning his law career
  • Enjoyed novels, painting, botany, and mountain climbing
  • Died in 1895

Cayley wrote a tremendous number of mathematical papers on subjects ranging from analytical geometry and matrix theory to transformation theory, partition theory, determinant theory, and the theory of invariants. He wrote so prolifically that "the massive Collected Mathematical Papers of Cayley contains 966 papers and fills thirteen...volumes averaging about 600 pages per volume.1" Many of these works were shorter articles appearing in periodicals such as the Quarterly Journal of Mathematics.

In 1857, Cayley devised an algebra on matrices by defining identity matrices for addition and multiplication, sums, products, and scalar multiplication. This algebra is associative and non-commutative. In fact, Cayley's matrix algebra, along with Hamilton's quaternions and Grassmann's ordered n-tuples, led the way for the gazillions of algebras or systems we study today, such as groups, rings, monoids, integral domains, fields, and vector spaces.

Other major milestones of Cayley's work are Cayley's Theorem and the Cayley-Hamilton Theorem, the former a theorem of group theory, the latter of matrix theory. I find it interesting to note that the Cayley-Hamilton theorem was not actually formally proved by Cayley. Rather, he was uninterested in formal proof (as are all mathematical realists) and simply gave a computational verification for matrices of order 2 while noting he had completed a similar verification for matrices of order 3. He concluded that it was "unnecessary to undertake a formal proof of the theorem in the general case of a matrix of arbitrary degree."3 It was Sylvester who carried these investigations much further in terms of rigorous proof.

1Eves, Howard. An Introduction to the History of Mathematics, 5th ed. 1983: CBS College Publishing.
2Cullen, Charles G. Matrices and Linear Transformations, 2nd ed. 1990: Dover Publications, Inc.
3Burton, David M. The History of Mathematics: an Introduction, 4th ed. 1999: McGraw Hill Companies, Inc.

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