Page 247 - Linear Algebra Done Right

Chapter 10. Trace and Determinant
238
Proof: Suppose T ∈L(V). By the polar decomposition (7.41), there
Another proof of this
corollary is suggested
in Exercise 24 in this is an isometry S ∈L(V) such that
√
∗
chapter. T = S T T.
Thus
√
|det T|=|det S| det T T
∗
√
= det T T,
∗
where the first equality follows from 10.34 and the second equality
follows from 10.35.
Suppose V is a real inner-product space and T ∈L(V) is invertible.
The det T is either positive or negative. A careful examination of the
proof of the corollary above can help us attach a geometric meaning
to whichever of these possibilities holds. To see this, first apply the
√
real spectral theorem (7.13) to the positive operator T T, getting an
∗
√
orthonormal basis (e 1 ,...,e n ) of V such that T Te j = λ j e j , where
∗
√
λ 1 ,...,λ n are the eigenvalues of T T, repeated according to multi-
∗
√
We are not formally plicity. Because each λ j is positive, T T never reverses direction.
∗
defining the phrase Now consider the polar decomposition
“reverses direction”
√
∗
because these T = S T T,
comments are meant to √
where S ∈L(V) is an isometry. Then det T = (det S)(det T T). Thus
∗
be an intuitive aid to
whether det T is positive or negative depends on whether det S is pos-
our understanding, not
itive or negative. As we saw earlier, this depends on whether the space
rigorous mathematics.
on which S reverses direction has even or odd dimension. Because
T is the product of S and an operator that never reverses direction
√
(namely, T T), we can reasonably say that whether det T is positive
∗
or negative depends on whether T reverses vectors an even or an odd
number of times.
Now we turn to the question of volume, where we will consider only
n
the real inner-product space R (with its standard inner product). We
n
would like to assign to each subset Ω of R its n-dimensional volume,
denoted volume Ω (when n = 2, this is usually called area instead of
volume). We begin with cubes, where we have a good intuitive notion of
n
volume. The cube in R with side length r and vertex (x 1 ,...,x n ) ∈ R n
is the set