Mathematical Tools for Physics

7—Operators and Matrices 188

can always picture it as a piece of a square thatisat the origin. The shaded square that is 1 / 16 the area of the
big square goes over to a parallelogram that’s 1 / 16 the area of the big parallelogram. Same ratio.

An arbitrary shape can be divided into a lot of squares. That’s how you do an integral. The image of
the whole area is distorted, but it retains the fact that a square that was inside the original area will become a
parallelogram that is inside the new area. In the limit as the number of squares goes to infinity you still maintain
the same ratio of areas as for the single original square.

7.8 Eigenvalues and Eigenvectors
There is a particularly important basis for an operator, the basis in which the components form a diagonal matrix.
Such a basis almost always exists, and it’s easy to seefrom the definitionas usual just what this basis must be.

f(~ei) =

∑N

k=1

fki~ek

To be diagonal simply means thatfki= 0for alli 6 =k, and that in turn means that all but one term in the sum
disappears. This defining equation reduces to

f(~ei) =fii~ei (with no sum this time) (27)