Mathematical Tools for Physics

7—Operators and Matrices 195

which determines the first column of the matrix(S), then~e′ 2 determines the second column.

(S) =

(

2 0. 5

0 .5 2

)

then (S)−^1 =

1

3. 75

(

2 − 0. 5

− 0 .5 2

)

Eigenvectors
In defining eigenvalues and eigenvectors I pointed out the utility of having a basis in which the components of
an operator form a diagonal matrix. Finding the non-zero solutions to Eq. ( 28 ) is then the way to find the basis
in which this holds. Now I’ve spent time showing that you can find a matrix in a new basis by using a similarity
transformation. Is there a relationship between these two subjects? Another way to ask the question: I’ve solved
the problem to find all the eigenvectors and eigenvalues, so what is the similarity transformation that accomplishes
the change of basis (and why do I need to know it if I already know that the transformed, diagonal matrix is just
the set of eigenvalues, and I already know them.)
For the last question, the simplest answer is that youdon’tneed to know the explicit transformation once
you already know the answer. It is however useful to know that it exists and how to construct it.If it exists — I’ll
come back to that presently. Certain manipulations are more easily done in terms of similarity transformations,
so you ought to know how they are constructed, especially because almost all the work in constructing them is
done when you’ve found the eigenvectors.
The equation ( 33 ) tells you the answer. Suppose that you want the transformed matrix to be diagonal.
That means thatf 12 ′ = 0andf 21 ′ = 0. Write out the first column of the product on the right.
(
f 11 f 12
f 21 f 22

)(

S 11 S 12

S 21 S 22

)

−→

(

f 11 f 12

f 21 f 22

)(

S 11

S 21

)

This equals the first column on the left of the same equation

f 11 ′

(

S 11

S 21

)

This is the eigenvector equation that you’ve supposedly already solved. The first column of the component matrix
of the similarity transformation is simply the set of components of the first eigenvector. When you write out the
second column of Eq. ( 33 ) you’ll see that it’s the defining equation for the second eigenvector. You already know
these, so you can immediately write down the matrix for the similarity transformation.