Elementary Abstract Algebra- Examples and Applications, 2019a

350 CHAPTER 8 SIGMA NOTATION
is a rotation matrix whose (j, k) entry is denoted by r jk . Then the equality
in Proposition 8.6.16 applied to matrix R becomes:
∑
ɛ jkl r ij r kl = ∑ ɛ jkl r ji r kl ,
j,k,l j,k,l
which implies (by rearranging terms)
⎛ ⎞
∑
r ij
⎝ ∑ ɛ jkl r kl
⎠ = ∑
j k,l
j
⎛ ⎞
r ji
⎝ ∑ ɛ jkl r kl
⎠ .
k,l
The expressions in parentheses on the left and right are identical. So let’s
define:
z j := ∑ ɛ jkl r kl ,
k,l
and we can replace the parenthetical expressions in our equality by z j :
∑
r ij z j = ∑ r ji z j .
j
j
Rewriting this in matrix notation gives Rz = R T z. Using the fact that
R T = R −1 (see Proposition 8.6.10) and a series of algebraic manipulations,
we find:
Now, there are two cases to consider:
Rz = R −1 z ⇒ Rz − R −1 z =0
⇒ R 2 z − Iz =0
⇒ (R + I)(R − I)z =0.
• Inthecasewhere(R−I)z ≠0,thenitmustbetruethaty := (R−I)z
is a nonzero vector which satisfies (R + I)y =0. Thisimpliesthaty
is an eigenvector of R with eigenvalue −1. Now since R is 3 × 3, it
must have 3 eigenvalues in total. Let λ 1 and λ 2 be the 2 remaining
eigenvalues. We know from linear algebra that the product of the
eigenvalues is equal to the determinant of R, which is equal to 1 by
Proposition 8.6.10. This imples that −1 · λ 1 · λ 2 =1orλ 1 · λ 2 = −1.
Now, the λ’s could be complex, or they could be real. If complex,
then they must be complex conjugates of each other (since R is a real