8.3 MATRICES
8.2.1 Properties of linear operatorsIfxis a vector andAandBare two linear operators then it follows that
(A+B)x=Ax+Bx,
(λA)x=λ(Ax),(AB)x=A(Bx),where in the last equality we see that the action of two linear operators in
succession is associative. The product of two linear operators is not in general
commutative, however, so that in generalABx=BAx. In an obvious way we
define the null (or zero) and identity operators by
Ox= 0 and Ix=x,for any vectorxin our vector space. Two operatorsA andB are equal if
Ax=Bxfor all vectorsx. Finally, if there exists an operatorA−^1 such that
AA−^1 =A−^1 A=IthenA−^1 is theinverseofA. Some linear operators do not possess an inverse
and are calledsingular, whilst those operators that do have an inverse are termed
non-singular.
8.3 MatricesWe have seen that in a particular basiseiboth vectors and linear operators
can be described in terms of their components with respect to the basis. These
components may be displayed as an array of numbers called amatrix.Ingeneral,
if a linear operatorAtransforms vectors from anN-dimensional vector space,
for which we choose a basisej,j=1, 2 ,...,N, into vectors belonging to an
M-dimensional vector space, with basisfi,i=1, 2 ,...,M, then we may represent
the operatorAby the matrix
A=
A 11 A 12 ... A 1 N
A 21 A 22 ... A 2 N
..
...
...
...
.
AM 1 AM 2 ... AMN
. (8.25)
Thematrix elementsAijare the components of the linear operator with respect
to the basesejandfi; the componentAijof the linear operator appears in the
ith row andjth column of the matrix. The array hasMrows andNcolumns
and is thus called anM×Nmatrix. If the dimensions of the two vector spaces
are the same, i.e.M=N(for example, if they are the same vector space) then we
may representAby anN×Norsquarematrix oforderN. The componentAij,
which in general may be complex, is also denoted by (A)ij.