4-5. 고유값 분해와 선형변환

본 글은 주재걸교수님의 인공지능을 위한 선형대수 강의를 듣고 정리한 내용입니다.

 

Eigendecomposition

• A가 diagonalizable하면, 𝐷=𝑉−1𝐴𝑉.
• 위 식은 𝐴=𝑉𝐷𝑉−1 로 쓸 수 있다. 이를 eigendecomposition of 𝐴 라고 부른다.
  • V : invertible(역행렬 존재)
• 𝐴 being diagonalizable is equivalent to 𝐴 having eigendecomposition
• Linear Transformation 𝑇(x)=𝐴x
• 𝑇(x) = 𝐴x = 𝑉𝐷𝑉−1x = 𝑉(𝐷(𝑉−1x))

1. Change of Basis

• Let y=𝑉−1x. Then, 𝑉y=x
• y is a new coordinate of x with respect to a new basis of eigenvectors {𝐯1,𝐯2}.

 

2. Element-wise Scaling

• 𝑇(x)=𝑉(𝐷(𝑉−1x))=𝑉(𝐷y)
• Let 𝑧=𝐷y. This computation is a simple Element-wise scaling of y.

 

3. Dimension-wise Scaling

4. Back to Original Basis

• 𝑇(x)=𝑉(𝐷y)=𝑉𝑧
• 𝑧 is still a coordinate based on the new basis {𝐯1,𝐯2}
• 𝑉𝑧 converts 𝑧 to another coordinates based on the original standard basis.

Linear Transformation via A^k

 

출처: https://www.edwith.org/ai251