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Homogeneous transformations and the history of matrix algebra

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  • Homogeneous transformations and the history of matrix algebra

    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"
    ( )
    [...] 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