7—Operators and Matrices 156What is the area of this parallelogram?
I’ll ask a more general question. (It isn’t really, but it looks like it.) Start withanyregion in theplane, and say it has areaA 1. The operator takes all the vectors ending in this area into some new area
of a sizeA 2 , probably different from the original. What is the ratio of the new area to the old one?
A 2 /A 1. How much does this transformation stretch or squeeze the area? What isn’t instantly obvious
is that this ratio of areas depends on the operatoralone,and not on how you chose the initial region
to be transformed. If you accept this for the moment, then you see that the question in the previous
paragraph, which started with the unit square and asked for the area into which it transformed, is the
same question as finding the ratio of the two more general areas. (Or the ratio of two volumes in three
dimensions.) See the end of the next section for a proof.
This ratio is called the determinant of the operator.The first example is the simplest. Rotations in the plane,Rα. Because rotations leave area
unchanged, this determinant is one. For almost any other example you have to do some work. Use the
component form to do the computation. The basis vector~e 1 is transformed into the vectorf 11 ~e 1 +f 21 ~e 2
with a similar expression for the image of~e 2. You can use the cross product to compute the area of
the parallelogram that these define. For another way, see problem7.3. This is
(f 11 ~e 1 +f 21 ~e 2
)
×
(
f 12 ~e 1 +f 22 ~e 2
)
=
(
f 11 f 22 −f 21 f 12
)
~e 3 (7.38)
The product in parentheses is the determinant of the transformation.det(f) =f 11 f 22 −f 21 f 12 (7.39)
What if I had picked a different basis, maybe even one that isn’t orthonormal? From the definition of
the determinant it is a property of the operator and not of the particular basis and components you
use to describe it, so you must get the same answer. But will the answer be the same simple formula
(7.39) if I pick a different basis? Nowthat’sa legitimate question. The answer is yes, and that fact
will come out of the general computation of the determinant in a moment. [What is the determinant
of Eq. (7.13)?]
det> 0
det< 0
The determinant can be either positive or negative. That tells you more than simply how the
transformation alters the area; it tells you whether it changes theorientationof the area. If you place
a counterclockwise loop in the original area, does it remain counterclockwise in the image or is it
reversed? In three dimensions, the corresponding plus or minus sign for the determinant says that
you’re changing from a right-handed set of vectors to a left-handed one. What does that mean? Make