7—Operators and Matrices 172This function satisfies the same linearity equations as Eq. ( 1 ). When you multiply~ωby a constant, the
output,~Lis multiplied by the same constant. When you add two~ω’s together as the argument, the properties
of the cross product and of the integral guarantee that the corresponding~L’s are added.
I(c~ω) =cI(~ω), and I(~ω 1 +~ω 2 ) =I(~ω 1 ) +I(~ω 2 )This functionIis called the “inertia operator” or more commonly the “inertia tensor.” It’s not simply multiplica-
tion by a scalar, so the rule that appears in an introductory course in mechanics (~L=I~ω) is valid only in special
cases, for example those with enough symmetry.
Note: Iis not a vector andL~ is not a function. ~Lis the output of the functionI when you feed it the
argument~ω. This is the same sort of observation that I made in section6.3under “Function Spaces.”
If an electromagnetic wave passes through a crystal, the electric field will push the electrons around, and the
bigger the electric field, the greater the distance that the electrons will be pushed. They may not be pushed in the
same direction as the electric field however, as the nature of the crystal can make it easier to push the electrons
in one direction than in another. The relation between the applied field and the average electron displacement is
a function that (for moderate size fields) obeys the same linearity relation that the two previous functions do.
P~=α(E~)P~is the electric dipole moment density andE~is the applied electric field. The functionαis called the polarizability.
If you have a mass attached to six springs that are in turn attached to six walls, the mass
will come to equilibrium somewhere. Now push on this mass with another (not too large) force.
The mass will move, but will it move in the direction that you push it? If the six springs are
all the same it will, but if they’re not then the displacement will be more in the direction of the
weaker springs. The displacement,d~, will still however depend linearly on the applied force,F~.
7.2 Definition of an Operator
An operator, also called a linear transformation, is a particular type of function. It is first of all,
a vector valued function of a vector variable. Second, it is linear; that is, ifAis such a function
thenA(~v)is a vector, and
A(α~v 1 +β~v 2 ) =αA(~v 1 ) +βA(~v 2 ). (4)Thedomainis the set of variables on which the operator is defined. Therangeis the set of all values put out by
the function.