Bounding the spectral norm by row and column sums

ProofIntermediate⏱ 30 minFoundationsoperator normholderinterpolation

Problem

For $A \in \R^{m\times n}$ define

R = \max_i \sum_j |A_{ij}| \quad (\text{max row sum}), \qquad C = \max_j \sum_i |A_{ij}| \quad (\text{max column sum}).

Prove that $\|A\|_{\infty\to\infty} = R$ , that $\|A\|_{1\to1} = C$ , and that

\|A\|_{2\to2} \le \sqrt{RC}.

Verify the last bound is tight for the all-ones matrix.

Write a complete argument. Every inequality you use should be one you could state precisely.

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