Question: Let X = [x 1 , . . . , x m ] R n,m , and p [1,+ ]. We consider the

Let X = [x₁, . . . , x_m] ∈ R^n,m, and p ∈ [1,+ ∞]. We consider the problem

T. Op (X) = max u max ||XTu||puTu= 1.

If the data is centered, that is, X₁ = 0, the above amounts of finding a direction of largest “deviation” from the origin, where deviation is measured using the l_p-norm.

1. Is Φ_p a (matrix) norm?

2. Solve the problem for p = 2. Find an optimal u.

3. Solve the problem for p = ∞. Find an optimal u.

T. Op (X) = max u max ||XTu||puTu= 1.

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