7—Operators and Matrices 152Why is this called the parallel axis theorem when you’re translating a point (the origin) and not an
axis? Probably because this was originally stated for the moment of inertia alone and not for the whole
tensor. In that case you have only an axis to deal with.
Components of the Derivative
The set of all polynomials inxhaving degree≤ 2 forms a vector space. There are three independent
vectors that I can choose to be 1 ,x, andx^2. Differentiation is a linear operator on this space because
the derivative of a sum is the sum of the derivatives and the derivative of a constant times a function
is the constant times the derivative of the function. With this basis I’ll compute the components of
d/dx. Start the indexing for the basis from zero instead of one because it will cause less confusion
between powers and subscripts.
~e 0 = 1, ~e 1 =x, ~e 2 =x^2
By the definition of the components of an operator — I’ll call this oneD,
D(~e 0 ) =
d
dx
1 = 0, D(~e 1 ) =
d
dx
x= 1 =~e 0 , D(~e 2 ) =
d
dx
x^2 = 2x= 2~e 1
These define the three columns of the matrix.
(D) =
0 1 0
0 0 2
0 0 0
check: dx
2dx
= 2xis
0 1 0
0 0 2
0 0 0
0
0
1
=
0
2
0
There’s nothing here about the basis being orthonormal. It isn’t.
7.4 Matrix Multiplication
How do you multiply two matrices? There’s a rule for doing it, but where does it come from?
The composition of two functions means you first apply one function then the other, so
h=f◦g means h(~v) =f
(
g(~v)
)
(7.24)
I’m assuming that these are vector-valued functions of a vector variable, but this is the general definition
of composition anyway. Iffandgare linear, does it follow thehis? Yes, just check:
h(c~v) =f
(
g(c~v)
)
=f
(
cg(~v)
)
=cf
(
g(~v)
)
, and
h(~v 1 +~v 2 ) =f
(
g(~v 1 +~v 2 )
)
=f
(
g(~v 1 ) +g(~v 2 )
)
=f
(
g(~v 1 )
)
+f
(
g(~v 2 )
)
What are the components ofh? Again, use the definition and plug in.
h(~ei) =
∑
khki~ek=f
(
g(~ei)
)
=f
(∑
jgji~ej
)
=
∑
jgjif
(
~ej
)
=
∑
jgji
∑
kfkj~ek (7.25)
and now all there is to do is to equate the corresponding coefficients of~ek.
hki=
∑
jgjifkj or more conventionally hki=
∑
jfkjgji (7.26)
This is in the standard form for matrix multiplication, recalling the subscripts are ordered asfrcfor
row-column.