1. Let V be a Hilbert space. Let a ∈ V be a given vector. The function Ax := ha,xi can be viewed as a linear transformation from V to R. Find the operator norm kAk.
2. Let f1,f2,…,fn are differentiable functions from V 7→ R with V a Hilbert space. Define F : V 7→ Rn by
f1(x)
f2(x) F(x) = … , ∀ x ∈ V.
fn(x)
Prove that
3. Find ∇f(x) and ∇2f(x).
, where A ∈ Rm×n, b ∈ Rm, and λ > 0 are given.
(b) f(X) = bTXc, where X ∈ Rn×n and b,c ∈ Rn.
(c) f(x) = xTAx, where x ∈ Rn, and A ∈ Rn×n is non-symmetric.
(d) f(X) = bTXTXc, where X ∈ Rn×n and b,c ∈ Rn.
(e) f(X) = trace(XAXB), where X,A,B ∈ Rn×n.
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