7—Operators and Matrices 183Matrix multiplication is just the component representation of the composition of two functions, Eq. ( 21 ).
There’s nothing here that restricts this to three dimensions.
Composition of Rotations
In the first example, rotating vectors in the plane, the operator that rotates every vector by the angleα has
components
(
Rα
)
=
(
cosα −sinα
sinα cosα)
What happens if you do two such transformations, one byαand one byβ? The result better be a total rotation
byα+β. One function,Rβis followed by the second functionRαand the composition is
Rα+β=RαRβThis is mirrored in the components of these operators, so the matrices must obey the same equation.
(
cos(α+β) −sin(α+β)
sin(α+β) cos(α+β))
=
(
cosα −sinα
sinα cosα)(
cosβ −sinβ
sinβ cosβ)
Multiply the matrices on the right to get
(
cosαcosβ−sinαsinβ −cosαsinβ−sinαcosβ
sinαcosβ+ cosαsinβ cosαcosβ−sinαsinβ)
(23)
The respective components must agree, so this gives an immediate derivation of the formulas for the sine and
cosine of the sum of two angles. Cf. Eq. (3.8)
7.5 Inverses
The simplest operator is the one that does nothing. f(~v) =~vfor all values of the vector~v. This implies that
f(~e 1 ) =~e 1 and similarly for all the other elements of the basis, so the matrix of its components is diagonal. The
2 × 2 matrix is explicitly the identity matrix
(I) =
(
1 0
0 1
)
or in index notation δij={
1 (ifi=j)
0 (ifi 6 =j)