Mathematical Tools for Physics

7—Operators and Matrices 184

and the index notation is completely general, not depending on whether you’re dealing with two dimensions or
many more. Unfortunately the words “inertia” and “identity” both start with the letter “I” and this symbol is
used for both operators. Live with it. Theδsymbol in this equation is the Kronecker delta — very handy.
Theinverseof an operator is defined in terms of Eq. ( 20 ), the composition of functions. If the composition
of two functions takes you to the identity operator, one function is said to be the inverse of the other. This is no
different from the way you look at ordinary real valued functions. The exponential and the logarithm are inverse
to each other because*
ln(ex) =x for allx.

For the rotation operator, Eq. ( 10 ), the inverse is obviously going to be rotation by the same angle in the opposite
direction.
RαR−α=I

Because the matrix components of these operators mirror the the original operators, this equation must also hold
for the corresponding components, as in Eqs. ( 22 ) and ( 23 ). Setβ=−αin ( 23 ) and you get the identity matrix.
In an equation such as Eq. ( 7 ), or its component form Eqs. ( 8 ) or ( 9 ), if you want to solve for the vector
~u, you are asking for the inverse of the functionf.

~u=f(~v) implies ~v=f−^1 (~u)

The translation of these equations into components is Eq. ( 9 )

(

u 1

u 2

)

=

(

f 11 f 12

f 21 f 22

)(

v 1

v 2

)

which implies

1

f 11 f 22 −f 12 f 21

(

f 22 −f 12

−f 21 f 11

)(

u 1

u 2

)

=

(

v 1

v 2

)

The verification that these are the components of the inverse is no more than simply multiplying the two matrices
and seeing that you get the identity matrix.

• The reverse,elnx works only for positivex, unless you recall that the logarithm of a negative number is
  complex. Then it works there too. This sort of question doesn’t occur with finite dimensional matrices.