Page 161 - Linear Algebra Done Right

Isometries
(g)
∗
∗
(h)
SS = I;
∗
(S e 1 ,...,S e n ) is orthonormal whenever (e 1 ,...,e n ) is an or-
(i) S u, S v = u, v for all u, v ∈ V; 149
∗
∗
thonormal list of vectors in V;
(j) there exists an orthonormal basis (e 1 ,...,e n ) of V such that
(S e 1 ,...,S e n ) is orthonormal.
∗
∗
Proof: First suppose that (a) holds. If V is a real inner-product
space, then for every u, v ∈ V we have
2
2
Su, Sv = ( Su + Sv −HSu − Sv )/4
2
2
= ( S(u + v) −HS(u − v) )/4
2
2
= ( u + v −Hu − v )/4
= u, v ,
where the first equality comes from Exercise 6 in Chapter 6, the second
equality comes from the linearity of S, the third equality holds because
S is an isometry, and the last equality again comes from Exercise 6 in
Chapter 6. If V is a complex inner-product space, then use Exercise 7
in Chapter 6 instead of Exercise 6 to obtain the same conclusion. In
either case, we see that (a) implies (b).
Now suppose that (b) holds. Then
∗
(S S − I)u, v = Su, Sv −Pu, v
= 0

for every u, v ∈ V. Taking v = (S S − I)u, we see that S S − I = 0.
∗
∗
Hence S S = I, proving that (b) implies (c).
∗
Now suppose that (c) holds. Suppose (e 1 ,...,e n ) is an orthonormal
list of vectors in V. Then
∗
Se j ,Se k = S Se j ,e k
= e j ,e k .

Hence (Se 1 ,...,Se n ) is orthonormal, proving that (c) implies (d).
Obviously (d) implies (e).
Now suppose (e) holds. Let (e 1 ,...,e n ) be an orthonormal basis of V
such that (Se 1 ,...,Se n ) is orthonormal. If v ∈ V, then