(2) Find the singular value decomposition of A. In other words, factorize A = , where U ∈ M₂(R) and V ∈ M₃(R) are orthogonal matrices and Σ ∈ (R) is of the form Σ = , λ₁ ≥ λ₂ ≥ 0

Notation: R is the set of real numbers, and C is the set of complex nunbers. If F=R or C, denote by (F) the n n matrices with entries in F. If A ∈ (F),denote by ∈ (F) the transpose of A. Denote by the n x n identity matrix and the n x n zero matrix. Problem 1 :Let i = √-1∈ C be a root of X² + 1. Let v₁ = (1,0,-1), v₂ = (1 + i,1 − i, 1), v₃ = (i, i, 1).Show that {v₁, v₂, v₃} is a basis of C³ and express the vector v₄ = (1,0,1) as a linear combination of v₁, v₂ and v₃, namely find α₁, α₂, α₃ ∈ C such that v₄ = α₁v₁ + α₂v₂ + α₃v₃.

題組內容

(2) Find the singular value decomposition of A. In other words, factorize A = , where U ∈ M₂(R) and V ∈ M₃(R) are orthogonal matrices and Σ ∈ (R) is of the form

Σ = , λ₁ ≥ λ₂ ≥ 0

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