4-4. 대각화

본 글은 주재걸교수님의 인공지능을 위한 선형대수 강의를 듣고 정리한 내용입니다.

 

Diagonalization

• a given square matrix 𝐴 ∈ ℝ𝑛×𝑛 into a diagonal matrix via the following form

• 𝑉 ∈ ℝ𝑛×𝑛 is an invertible matrix and 𝐷 ∈ ℝ𝑛×𝑛 is a diagonal matrix. This is called a diagonalization of 𝐴.
  • V는 A와 같은 dimension의 정사각행렬
• 𝐷 = 𝑉^(−1)𝐴𝑉 ⟹ 𝑉𝐷 = 𝐴𝑉

𝐴𝑉 = 𝑉𝐷 ⟺ [𝐴𝐯1 𝐴𝐯2 ⋯ 𝐴𝐯𝑛] = [𝜆1𝐯1 𝜆2𝐯2 ⋯ 𝜆𝑛𝐯𝑛] => 𝐴𝐯1 = 𝜆1𝐯1, 𝐴𝐯2 = 𝜆2𝐯2,  …,  𝐴𝐯𝑛 = 𝜆𝑛𝐯𝑛

∴ 𝐯1, 𝐯2, …, 𝐯𝑛 should be eigenvectors and 𝜆1, 𝜆2, …, 𝜆𝑛 should be eigenvalues.

 

 

• 𝑉 to be invertible, 𝑉 should be a square matrix in ℝ𝑛×𝑛 , and 𝑉 should have 𝑛 linearly independent columns.
• Recall columns of 𝑉 are eigenvectors. Hence, 𝐴 should have 𝑛 linearly independent eigenvectors.
• • It is not always the case, but if it is, 𝐴 is diagonalizable.

 

 

출처: https://www.edwith.org/ai251