68 Cayley, Arthur

It is based on the idea that the volume of a deck of

cards, for example, does not change even if the deck is

skewed. A close examination of

VOLUME

explains why

Cavalieri’s principle is true.

Cayley, Arthur (1821–1895) British Matrix theory,

Geometry, Abstract algebra, Analysis Born on August

16, 1821, in Richmond, England, Arthur Cayley is

remembered as a prolific writer, having produced 967

papers in all, covering nearly every aspect of modern

mathematics. His most significant work, Memoir on the

Theory of Matrices (1858) established the new field of

matrices and

MATRIX

algebra. Cayley studied abstract

groups and was the first to study geometry in n-dimen-

sional space with na number greater than three.

Cayley demonstrated a great aptitude for mathemat-

ics as a child. A schoolteacher recognized his talent and

encouraged Cayley’s father to allow him to pursue stud-

ies in mathematics rather than leave school and enter the

family retail business. Cayley attended Trinity College,

Cambridge, and graduated in 1842. Unable to find an

academic position in mathematics, Cayley pursued a law

degree and practiced law for 14 years. During this time,

however, Cayley actively studied mathematics and

published over 250 research papers. In 1863 he was

finally appointed a professorship in mathematics at

Cambridge.

In 1854, while working as a lawyer, Cayley wrote

“On the Theory of Groups Depending on the Symbolic

Equation θn= 1” and other significant papers that

defined, for the first time, the notion of an abstract

GROUP

. At that time, the only known groups were

PER

-

MUTATION

groups, but Cayley realized that the mathe-

matical principles behind these structures also applied

to matrices, number systems, and geometric transforma-

tions. This work allowed Cayley to begin analyzing the

geometry of higher-dimensional space. This, in turn,

coupled with his newly developed matrix algebra, pro-

vided the foundation for the theory of quantum

mechanics, as developed by Werner Heisenberg in 1925.

Cayley was elected president of the British Associa-

tion for the Advancement of Science in 1883. From the

years 1889 to 1895, Cayley’s entire mathematical out-

put was collated into one 13-volume work, The Col-

lected Mathematical Papers of Arthur Cayley. This

project was supervised by Cayley himself until he died

on January 26, 1895, at which point only seven vol-

umes had been produced. The remaining six volumes

were edited by A. R. Forsyth.

See also C

AYLEY

-H

AMILTON THEOREM

.

Cayley-Hamilton theorem English mathematician

A

RTHUR

C

AYLEY

(1821–95) and Irish mathematician

S

IR

W

ILLIAM

R

OWAN

H

AMILTON

(1805–65) noted that

any square

MATRIX

satisfies some polynomial equa-

tion. To see this, first note that the set of all square n×

nmatrices with real entries forms a

VECTOR SPACE

over the real numbers. For example, the set of 2 ×2

square matrices is a four-dimensional vector space

with basis elements:

and any 2 ×2 matrix is indeed a linear

combination of these four linearly independent matrices:

In general, the set of all n×nsquare matrices is an

n2-dimensional vector space. In particular then, for

any n×nmatrix A, the n2+ 1 matrices I, A, A2, A3,

…, An

2

must be linearly dependent, that is, there is a

linear combination of these elements that yields the

zero matrix:

c0I+ c1A+ c2A2+ … + cn

2

An

2

= 0

for some numbers c0, …, cn

2

. This shows:

Any n×nsquare matrix satisfies a polynomial

equation of degree at most n2.

Cayley and Hamilton went further and proved that

any square matrix satisfies its own “characteristic

polynomial”:

Let Abe an n×nsquare matrix and set xto be

a variable. Subtract xfrom each diagonal entry

of Aand compute the

DETERMINANT

of the

resulting matrix. This yields a polynomial in x