Let A ∈ℂ^p ×q. Show that (10.13) A=max{|⟨A 𝐱, 𝐲⟩|: 𝐱 ∈ℂ^q, 𝐲 ∈ℂ^p., and .𝐱=𝐲=1}.

Question

Accepted Answer

Step 1:
Define the operator norm of a matrix $A in mathbb{C}^{p times q}$, which is given by
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Let $A \in \mathbb{C}^{p \times q}$. Show that (10.13) $\|A\|=\max \left\{|\langle A \mathbf{x}, \mathbf{y}\rangle|: \mathbf{x} \in \mathbb{C}^q, \mathbf{y} \in \mathbb{C}^p\right.$, and $\left.\|\mathbf{x}\|=\|\mathbf{y}\|=1\right\}$.