View Full Version : Homogeneous transformations and the history of matrix algebra

Paolo De Leva
07-07-2005, 02:29 AM
Dear subscribers,

I believe the following text is quite an interesting piece of
history. It is interesting to know that matrices were developed just for the
purpose of performing linear (i.e. homogeneous) transformations and
composite linear transformations. It was done by Arthur Cayley in the
nineteenth century, before we were born, before electronic computers were
developed and before computer graphics could use that wonderful mathematical
The need for transformation composition led to the definition of
matrix multiplication.
Of course, 4x4 "general" transformation matrices to perform, in a 3D
space, combined homogeneous and not-homogeneous transformations such as
roto-translation were probably developed later. Does someone of you know who
and when introduced them?

From "A Brief History of Linear Algebra and Matrix Theory"
(http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html )
[...] For matrix algebra to fruitfully develop one needed both
proper notation and the proper definition of matrix multiplication. Both
needs were met at about the same time and in the same place. In 1848 in
England, J.J. Sylvester first introduced the term ''matrix,'' which was the
Latin word for womb, as a name for an array of numbers. Matrix algebra was
nurtured by the work of Arthur Cayley in 1855. Cayley studied compositions
of linear transformations and was led to define matrix multiplication so
that the matrix of coefficients for the composite transformation ST is the
product of the matrix for S times the matrix for T. He went on to study the
algebra of these compositions including matrix inverses. The famous
Cayley-Hamilton theorem which asserts that a square matrix is a root of its
characteristic polynomial was given by Cayley in his 1858 Memoir on the
Theory of Matrices. The use of a single letter A to represent a matrix was
crucial to the development of matrix algebra. [...]

With kind regards,

Paolo de Leva
Sport Biomechanics
University Institute of Motor Sciences
Rome, Italy