Mathematical Tools for Physics

7—Operators and Matrices 174

The way to write a vector~vin components is

~v=vxˆx+vyˆy+vzˆz, or v 1 ~e 1 +v 2 ~e 2 +v 3 ~e 3 =

∑

k

vk~ek (5)

Once you’ve chosen a basis, you can find the three numbers that form the components of that vector. In
a similar way, I will define the components of an operator, only that will takenine numbers to do it (in three
dimensions). If you evaluate the effect of an operator on any one of the basis vectors, the output is a vector.
That’s part of the definition of the word operator. This output vector can itself be written in terms of this same
basis. The defining equation for the components of an operatorfis

f(~ei) =

∑^3

k=1

fki~ek (6)

For each input vector you have the three components of the output vector. Pay careful attention to this
equation! It is the defining equation for the entire subject of matrix theory, and everything in that subject comes
from this one innocuous looking equation. (And yes if you’re wondering, I wrote the indices in the correct order.)
Why?
Take an arbitrary input vector forf: ~u=f(~v). Both~uand~vare vectors, so I can write them in terms of
my chosen basis.

~u=

∑

k

uk~ek=f(~v) =f

(∑

i

vi~ei

)

=

∑

i

vif(~ei) (7)

The last equation is the result of the linearity property, Eq. ( 1 ), that I have assumed forf. I can pull the sum
and the numerical factorsviout in front of the function. Write it out and it’s clear:

f(v 1 ~e 1 +v 2 ~e 2 ) =f(v 1 ~e 1 ) +f(v 2 ~e 2 ) =v 1 f(~e 1 ) +v 2 f(~e 2 )

Now you see where the defining equation for operator components comes in. Eq. ( 7 ) is

∑

k

uk~ek=

∑

i

vi

∑

k

fki~ek