39 Matrices
This is the story of ‘extraordinary algebra’ – a revolution in mathematics which took
place in the middle of the 19th century. Mathematicians had played with blocks of
numbers for centuries, but the idea of treating blocks as a single number took off 150
years ago with a small group of mathematicians who recognized its potential.
Ordinary algebra is the traditional algebra in which symbols such as a, b, c, x
and y represent single numbers. Many people find this difficult to understand, but
for mathematicians it was a great step forward. In comparison, ‘extraordinary
algebra’ generated a seismic shift. For sophisticated applications this progress
from a one-dimensional algebra to a multiple dimensional algebra would prove
incredibly powerful.
Multiple dimensioned numbers
In ordinary algebra a might represent a number such as 7, and we would
write a = 7, but in matrix theory a matrix A would be a ‘multiple dimensioned
number’ for example the block
This matrix has three rows and four columns (it’s a ‘3 by 4’ matrix), but in
principle we can have matrices with any number of rows and columns – even a
giant ‘100 by 200’ matrix with 100 rows and 200 columns. A critical advantage of
matrix algebra is that we can think of vast arrays of numbers, such as a data set
in statistics, as a single entity. More than this, we can manipulate these blocks of
numbers simply and efficiently. If we want to add or multiply together all the
numbers in two data sets, each consisting of 1000 numbers, we don’t have to
perform 1000 calculations – we just have to perform one (adding or multiplying
the two matrices together).