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Matrix And It’s Application In Science
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For matrix algebra to fruitfully develop one needed both proper notation
and proper definition of matrix multiplication. Both needs were met at
about the same time in the same place. In 1848 in England, J.J Sylvester
first introduced the term “matrixâ€, which was the Latin word for “wombâ€
as a name for an array of numbers.
Matrix algebra was nurtured by
the work of Arthur Cayley in 1855. Cayley studied multiplication so that
the matrix of coefficient for the composite transformation ST is the
product of the matrix S times the matrix T. He went on to study the
algebra of these composition including matrix inverses. The famous
Cayley-Hamilton theorem which asserts that a square matrix is a root of
it’s characteristics’ polynomial was given by Cayley in his 1858 memoir
on the theory of matrices. The use of single letter “A to represent a
matrix was crucial to the development of matrix algebra. Early in the
development, the formular det(AB) = det (A) det(B) provided a connection
between matrix algebra and determinants. Cayley wrote “There would be
many things to say about this theory of matrices which should, it seems
to me, precede the theory of determinantsâ€.
Mathematicians also
attempted to develop for algebra of vectors but there was no natural
definition of the product of two vectors that held in arbitrary
dimensions. The first vector algebra that involved a non- commutative
vector product (that is V x W need not equal W x V) was proposed by
Hermann Grassman in his book – Ausedehnungslehre (1844). Grossmann’s
text also introduced the product of a column matrix and a row matrix,
which resulted in what is now called a simple or a rank one matrix. In
the late 19th century the American mathematical physicist, Willard Gibbs
published his famous treatise on vector analysis. In that treatise
Gibbs represented general matrices, which he called dyadics as sum of
simple matrices, which Gibbs called dyads. Later the physicist, P.A.M.
Dirac introduced the term “bracket†for what we now call the “scalar
product†of a “bar†(row) vector times a “ket†(column) vector and the
term “ket-bra†for the product of a ket times a bra, resulting in what
we now call a simple matrix, as above. Our convention of identifying
column matrices and vector was introduced by physicists in the 20th
century.
CHAPTER ONE -- [Total Page(s) 3]
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