Mathematical Tools for Physics - Department of Physics - University

7—Operators and Matrices 165

7.10 Change of Basis
In many problems in physics and mathematics, the correct choice of basis can enormously simplify a
problem. Sometimes the obvious choice of a basis turns out in the end not to be the best choice, and
you then face the question: Do you start over with a new basis, or can you use the work that you’ve
already done to transform everything into the new basis?
For linear transformations, this becomes the problem of computing the components of an operator
in a new basis in terms of its components in the old basis.
First: Review how to do this for vector components, something that ought to be easy to do. The
equation (7.5) defines the components with respect to a basis,any basis. If I have a second proposed
basis, then by the definition of the word basis, every vector in that second basis can be written as a

linear combination of the vectors in the first basis. I’ll call the vectors in the first basis,~eiand those in

the second basis~e′i, for example in the plane you could have

~e 1 =ˆx, ~e 2 =y,ˆ and ~e′ 1 = 2xˆ+ 0. 5 y, ~eˆ ′ 2 = 0. 5 ˆx+ 2yˆ (7.55)

Each vector~e′iis a linear combination* of the original basis vectors:

~e′i=S(~ei) =

∑

j

Sji~ej (7.56)

This follows the standard notation of Eq. (7.6); you have to put the indices in this order in order to
make the notation come out right in the end. One vector expressed in two different bases is still one
vector, so

~v=

∑

i

v′i~e′i=

∑

i

vi~ei

and I’m using the fairly standard notation ofv′ifor theithcomponent of the vector~vwith respect to

the second basis. Now insert the relation between the bases from the preceding equation (7.56).

~v=

∑

i

v′i

∑

j

Sji~ej=

∑

j

vj~ej

and this used the standard trick of changing the last dummy label of summation fromitojso that it

is easy to compare the components.

∑

i

Sjiv′i=vj or in matrix notation (S)(v′) = (v),=⇒ (v′) = (S)−^1 (v)

Similarity Transformations
Now use the definition of the components of an operator to get the components in the new basis.

f

(

~e′i

)

= =

∑

j

fji′~e′j

f

(∑

j

Sji~ej

)

=

∑

j

Sjif

(

~ej

)

=

∑

j

Sji

∑

k

fkj~ek=

∑

j

fji′

∑

k

Skj~ek

* There are two possible conventions here. You can write~e′i in terms of the~ei, calling the

coefficientsSji, or you can do the reverse and callthosecomponentsSji. [~ei=S(~e′i)] Naturally, both

conventions are in common use. The reverse convention will interchange the roles of the matricesS

andS−^1 in what follows.