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

tool.

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

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# Homogeneous transformations and the history of matrix algebra

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