## Arthur Cayley (1821 - 1895)

Born in England, Cayley spent the first eight years of his childhood in St. Petersburg, Russia, returning to London in 1828.

At school Cayley showed great skill in numerical calculations and displayed his propensity for advanced mathematics while studying at King’s College School in 1835.

Whilst here, Cayley’s mathematics teacher encouraged him to pursue his mathematical studies rather than follow his father’s wish to work in the family business.

In 1838 Cayley went to Trinity College, Cambridge. He became top student of his year graduating in 1842. He taught at Cambridge for four years having won a Fellowship and, during this period, published an impressive 28 papers.

Cayley left Cambridge in 1846 to study law in London, and was admitted to the bar in 1849. He practised law from 1849 until 1863, while writing over 250 mathematical papers in his spare time! In recognition of this mathematical work, he was elected to the Royal Society in 1852 and presented with its Royal Medal in 1859.

In 1863 he left the legal profession to accept the Sadleirian professorship in mathematics at Cambridge, a full-time mathematical post. In that same year he married Susan Moline, the daughter of a country banker.

He was an active supporter of women’s higher education and steered Newnham College, Cambridge during the 1880s.

Cayley was highly recognised throughout his lifetime, and at various times was president of the Cambridge Philosophical Society, the London Mathematical Society, the British Association for the Advancement of Science, the Royal Astronomical Society, and the British Association for the Advancement of Science In Mathematics.

## Cayley’s mathematics

A matrix is a table of ordered entries - usually numbers, but sometimes functions. Cayley was intrigued by the algebra of such objects, and he was the first to realise that they unified a number of contemporary areas of mathematics - permutations, geometric transformations, groups and even the fundamental ideas that underpin algebra itself. A simple example of a matrix A is:

A=\bigl({2\atop 0}{1\atop 3}\bigr)

The Cayley Hamilton Theorem says that the matrix A always obeys a particular equation - called its characteristic equation. In this case that is

A^2-5A+6I=0

where I is the identity matrix I=\bigl({1\atop 0}{0\atop 1}\bigr)

It also shows how to find higher powers of the matrix A and an inverse, if it exists.

If we multiply through the characteristic equation by A^n we find that:

A^{n+2}=5A^{n+1}-6A^n

So if A^n and A^{n+1} are known then so is A^{n+2}

If the inverse matrix A^{-1} exists we may multiply through the characteristic equation by it:

A-5I+6A^{-1}=0

so that: {A^{-1}}={1\over6}{(A-5I)}

Cayley also made important contributions to geometry, but he did so in an algebraic way. He sought to understand curves and surfaces by looking at the algebraic ways that they may be described, and then searching for invariants - things that don’t change when other things are changed.

Such invariants must represent intrinsic properties of the surfaces and curves. His ultimate goal was to classify surfaces and curves by their invariants - a huge task, and the subject of continuous efforts henceforth. Such an approach to geometry led to another spectacular advance - the geometry of dimensions higher than three.

Because it is algebraically based, it doesn’t matter how many dimensions there are - there are no ‘visual’ problems of dealing with such complexity.

For his pioneering work, his originality and sheer output he was awarded numerous honours. But his true honour was in the way that his work was the foundation of so many developments in other areas - especially physics.

His development of n-dimensional geometry has been applied in physics to the study of the space-time continuum. His work on matrices served as a foundation for quantum mechanics. His work on groups finds a natural place in the physics of atomic particles.