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-dimensional space with n a number greater than three.
Cayley demonstrated a great aptitude for mathematics as a child. A schoolteacher recognized his talent and
encouraged Cayley’s father to allow him to pursue studies 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 θ
n = 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 PERMUTATION groups, but Cayley realized that the mathematical principles behind these structures also applied
to matrices, number systems, and geometric transformations. This work allowed Cayley to begin analyzing the
geometry of higher-dimensional space. This, in turn,
coupled with his newly developed matrix algebra, provided the foundation for the theory of quantum
mechanics, as developed by Werner Heisenberg in 1925.
Cayley was elected president of the British Association for the Advancement of Science in 1883. From the
years 1889 to 1895, Cayley’s entire mathematical output was collated into one 13-volume work, The Collected Mathematical Papers of Arthur Cayley. This
project was supervised by Cayley himself until he died
on January 26, 1895, at which point only seven volumes had been produced. The remaining six volumes
were edited by A. R. Forsyth.
See also CAYLEY-HAMILTON THEOREM.
Cayley-Hamilton theorem English mathematician
ARTHUR CAYLEY (1821–95) and Irish mathematician
SIR WILLIAM ROWAN HAMILTON (1805–65) noted that
any square MATRIX satisfies some polynomial equation. To see this, first note that the set of all square n ×
n matrices 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 × n square matrices is an
n
2 -dimensional vector space. In particular then, for
any n × n matrix A, the n
2 + 1 matrices I, A, A
2 , A
3 ,
…, A
n
2 must be linearly dependent, that is, there is a
linear combination of these elements that yields the
zero matrix:
c 0 I + c 1 A + c 2 A
2 + … + c n 2A
n
2 = 0
for some numbers c 0 , …, c n 2 . This shows:
Any n × n square matrix satisfies a polynomial
equation of degree at most n
2
.
Cayley and Hamilton went further and proved that
any square matrix satisfies its own “characteristic
polynomial”:
Let A be an n × n square matrix and set x to be
a variable. Subtract x from each diagonal entry
of A and compute the DETERMINANT of the
resulting matrix. This yields a polynomial in x
A
a b
c d
a
b
c
d
=
=
+
+
+
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
A
a b
c d
=
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
,
,
, and
68 Cayley, Arthur
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-dimensional space with n a number greater than three.
Cayley demonstrated a great aptitude for mathematics as a child. A schoolteacher recognized his talent and
encouraged Cayley’s father to allow him to pursue studies 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 θ
n = 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 PERMUTATION groups, but Cayley realized that the mathematical principles behind these structures also applied
to matrices, number systems, and geometric transformations. This work allowed Cayley to begin analyzing the
geometry of higher-dimensional space. This, in turn,
coupled with his newly developed matrix algebra, provided the foundation for the theory of quantum
mechanics, as developed by Werner Heisenberg in 1925.
Cayley was elected president of the British Association for the Advancement of Science in 1883. From the
years 1889 to 1895, Cayley’s entire mathematical output was collated into one 13-volume work, The Collected Mathematical Papers of Arthur Cayley. This
project was supervised by Cayley himself until he died
on January 26, 1895, at which point only seven volumes had been produced. The remaining six volumes
were edited by A. R. Forsyth.
See also CAYLEY-HAMILTON THEOREM.
Cayley-Hamilton theorem English mathematician
ARTHUR CAYLEY (1821–95) and Irish mathematician
SIR WILLIAM ROWAN HAMILTON (1805–65) noted that
any square MATRIX satisfies some polynomial equation. To see this, first note that the set of all square n ×
n matrices 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 × n square matrices is an
n
2 -dimensional vector space. In particular then, for
any n × n matrix A, the n
2 + 1 matrices I, A, A
2 , A
3 ,
…, A
n
2 must be linearly dependent, that is, there is a
linear combination of these elements that yields the
zero matrix:
c 0 I + c 1 A + c 2 A
2 + … + c n 2A
n
2 = 0
for some numbers c 0 , …, c n 2 . This shows:
Any n × n square matrix satisfies a polynomial
equation of degree at most n
2
.
Cayley and Hamilton went further and proved that
any square matrix satisfies its own “characteristic
polynomial”:
Let A be an n × n square matrix and set x to be
a variable. Subtract x from each diagonal entry
of A and compute the DETERMINANT of the
resulting matrix. This yields a polynomial in x
A
a b
c d
a
b
c
d
=
=
+
+
+
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
A
a b
c d
=
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
,
,
, and
68 Cayley, Arthur