Page 160 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
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where we have used 6.17. Applying S to both sides of 7.33 gives
Sv = v, e 1 Se 1 +· · ·+Pv, e n Se n
= λ 1 v, e 1 e 1 + ··· + λ n v, e n e n .
The last equation, along with the equation |λ j |= 1, shows that
2
2
2
7.35 Sv =| v, e 1 | + ··· + | v, e n | .
Comparing 7.34 and 7.35 shows that v = Sv . In other words, S is
an isometry.
2
An isometry on a real For another example, let θ ∈ R. Then the operator on R of coun-
inner-product space is terclockwise rotation (centered at the origin) by an angle of θ is an
often called an isometry (you should find the matrix of this operator with respect to
2
orthogonal operator. the standard basis of R ).
An isometry on a If S ∈L(V) is an isometry, then S is injective (because if Sv = 0,
complex inner-product then v = Sv = 0, and hence v = 0). Thus every isometry is
space is often called a invertible (by 3.21).
unitary operator. We The next theorem provides several conditions that are equivalent
will use the term to being an isometry. These equivalences have several important in-
isometry so that our terpretations. In particular, the equivalence of (a) and (b) shows that
results can apply to an isometry preserves inner products. Because (a) implies (d), we see
both real and complex that if S is an isometry and (e 1 ,...,e n ) is an orthonormal basis of V,
inner-product spaces. then the columns of the matrix of S (with respect to this basis) are or-
thonormal; because (e) implies (a), we see that the converse also holds.
Because (a) is equivalent to conditions (i) and (j), we see that in the last
sentence we can replace “columns” with “rows”.

7.36 Theorem: Suppose S ∈L(V). Then the following are equiva-
lent:

(a) S is an isometry;
(b) Su, Sv = u, v for all u, v ∈ V;
(c) S S = I;
∗
(d) (Se 1 ,...,Se n ) is orthonormal whenever (e 1 ,...,e n ) is an ortho-
normal list of vectors in V;
(e) there exists an orthonormal basis (e 1 ,...,e n ) of V such that
(Se 1 ,...,Se n ) is orthonormal;
(f) S ∗ is an isometry;