Lecture 7 : Singular Value Decomposition
نویسنده
چکیده
∀ ` σ` ∈ R, σ` ≥ 0 (2) ∀ `, `′ 〈u`, u`′〉 = 〈v`, v`′〉 = δ(`, `′) (3) To prove this consider the matrix AA ∈ Rm×m. Set u` to be the `’th eigenvector of AA . By definition we have that AAu` = λ`u`. Since AA T is positive semidefinite we have λ` ≥ 0. Since AA is symmetric we have that ∀ `, `′ 〈u`, u`′〉 = δ(`, `′). Set σ` = √ λ` and v` = 1 σ` Au`. Now we can compute the following: 〈v`, v`′〉 = 1 σ2 ` u` AA u` = 1 σ2 ` λ`〈u`, u`′〉 = δ(`, `′)
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