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Exercise 1.11.7

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We know that $| {(AB)}_{ij} |^{2} \leq | row i of A |^{2} | column j of B |^{2}$ from Cachy-Schwarz inequality. Add up both sides over all $i$ and $j$ we have

\begin{array}{l} \sum_{i} \sum_{j} | {(AB)}_{ij} |^{2} & \leq \sum_{i} \sum_{j} | row i of A |^{2} | column j of B |^{2} \\ | AB |_{F}^{2} & = \sum_{i} | row i of A |^{2} \sum_{j} | column j of B |^{2} \\ = \sum_{i} | row i of A |^{2} | B |_{F}^{2} \\ = | A |_{F}^{2} | B |_{F}^{2} \end{array}

niuers

2020-03-20 00:00

Exercise 1.11.7

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