16. The group of regular matrices
Consider the totality Ln of all square, regular matrices with n-rows, the elements of which are real or complex numbers. According to the statements in Sections 14 and 15, Ln has the properties:
(1)
The product of two matrices of Ln is
again a matrix in Ln.
(2) The associative law applies.
(3) Ln contains
the unit matrix E with the property EA = AE
= A for every matrix in Ln.
(4) There exists in Ln
for every matrix A-1 exactly
one matrix with the property A-1A = AA-1
= E.
An aggregate for the elements of which an operation (in this case, it is matrix multiplication) is defined with the properties (1) - (4) above is called a group. Hence we can say: The aggregate Ln forms with respect to matrix multiplication a group.