7—Operators and Matrices 161The components ofM in this basis are
(
5 − 1
− 2 0
)
It doesn’t look at all the same, but it represents the same operator. Does this matrix have the same
determinant, using Eq. (7.39)?
Determinant of Composition
If you do one linear transformation followed by another one, that is the composition of the two functions,
each operator will then have its own determinant. What is the determinant of the composition? Let
the operators beFandG. One of them changes areas by a scale factordet(F)and the other ratio of
areas isdet(G). If you use the composition of the two functions,FGorGF, the overall ratio of areas
from the start to the finish will be the same:
det(FG) = det(F).det(G) = det(G).det(F) = det(GF) (7.48)
Recall that the determinant measures the ratio of areas for any input area, not just a square; it can be
a parallelogram. The overall ratio of the product of the individual ratios,det(F) det(G). The product
of these two numbers is the total ratio of a new area to the original area and it is independent of the
order ofFandG, so the determinant of the composition of the functions is also independent of order.
Now what about the statement that the definition of the determinant doesn’t depend on the
original area that you start with. To show this takes a couple of steps. First, start with a square that’s
not at the origin. You 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.
7.9 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=1fki~ek
* See section7.12.