Linear Algebra Done Right · Problem 7F.18

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms-orthonormal

Problem

Suppose

e_1, \ldots, e_n

is an orthonormal basis of

V

and

T \in \mathcal{L}(V, W)

. (a) Prove that

\max\{\|Te_1\|, \ldots, \|Te_n\|\} \leq \|T\| \leq (\|Te_1\|^2 + \cdots + \|Te_n\|^2)^{1/2}

. (b) Prove that

\|T\| = (\|Te_1\|^2 + \cdots + \|Te_n\|^2)^{1/2}

if and only if

\dim \operatorname{range} T \leq 1

. Here

e_1, \ldots, e_n

is an arbitrary orthonormal basis of

V

, not necessarily connected with a singular value decomposition of

T

. If

s_1, \ldots, s_n

is the list of singular values of

T

, then the right side of the inequality above equals

(s_1^2 + \cdots + s_n^2)^{1/2}

, as was shown in Exercise 11(a) in Section 7E.

Intuitive explanation

The operator norm $\|T\|$ measures the maximum stretch $T$ applies to any unit vector, while the images of basis vectors reveal how $T$ distributes its action across the space. Part (a) asks you to bound $\|T\|$ using these basis images: the left inequality follows because basis vectors are unit vectors, and the right uses Cauchy–Schwarz on how an arbitrary vector decomposes into basis elements. Part (b) explores when equality holds on the right—this happens precisely when $T$ "focuses" all its action in a single direction, meaning its range is at most one-dimensional.

Intuitive explanation

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