Mathematical Tools for Physics

7—Operators and Matrices 194

and I pulled the standard trick of changing the last dummy label of summation fromitojso that I can compare
the components more easily.

∑

i

Sjivi′=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

The final equation comes from the preceding line. The coefficients of~ek must agree on the two sides of the
equation. ∑

j

Sjifkj=

∑

j

fji′Skj

Now rearrange this in order to place the indices in their conventionalrow,columnorder.

∑

j

Skjfji′ =

∑

j

fkjSji

(

S 11 S 12

S 21 S 22

)(

f 11 ′ f 12 ′

f 21 ′ f 22 ′

)

=

(

f 11 f 12

f 21 f 22

)(

S 11 S 12

S 21 S 22

) (33)

In turn, this matrix equation is usually written in terms of the inverse matrix ofS,

(S)(f′) = (f)(S) is (f′) = (S)−^1 (f)(S)

and this is called a similarity transformation. For the example Eq. ( 31 ) this is

~e′ 1 = 2ˆx+ 0. 5 ˆy=S 11 ~e 1 +S 21 ~e 2