3-5. 그람-슈미트 직교화와 QR 분해

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Gram-Schmidt

Orthogonal 하지 않은 두 벡터에 대해 아래 식을 통해 orthogonal한 벡터를 만들어 준다.

v2 vector를 v1(=u1) 방향에 대하여 projection 해준다. => v2 vector

 

{𝐱1, 𝐱2, 𝐱3} is clearly linearly independent and thus is a basis for a subspace 𝑊 of ℝ4 . Construct an orthogonal basis for W

Solution

• Let 𝐯1 = 𝐱1 and 𝑊1 = Span {𝐱1} = Span{ 𝐯1}
• Let 𝐯2 be the vector produced by subtracting from 𝐱2 its projection onto the subspace 𝑊1

𝐯2 is the component of 𝐱2 orthogonal to 𝐱1, and {𝐯1, 𝐯2} is an orthogonal basis for the subspace 𝑊2 spanned by 𝐱1 and 𝐱2.

• Let 𝐯3 be the vector produced by subtracting from 𝐱3 its projection onto the subspace 𝑊2. Use the orthogonal basis {𝐯1, 𝐯2′} to compute this projection onto 𝑊2

QR Factorization

If 𝐴 is an 𝑚 × 𝑛 matrix with linearly independent columns, then 𝐴 can be factored as 𝐴 = 𝑄𝑅, where 𝑄 is an 𝑚 × 𝑛 matrix whose columns form an orthonormal basis for Col 𝐴 and 𝑅 is an 𝑛 × 𝑛 upper triangular invertible matrix with positive entries on its diagonal

 

 

출처: https://www.edwith.org/ai251