- Diagonal matrix: a matrix (not only a square matrix) in which the entries outside the main diagonal (aij, i=j; ↘) are all zero. The diagonal entries themselves may or may not be zero.
- Diagonalizable: a square matrix is diagonalizable if the matrix is similar to a diagonal matrix. (If there exists an invertible matrix P such that P-1AP is a diagonal matrix, the matrix A is diagonalizable.)
- Diagonalization: the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map.
If a matrix A can be diagonalized (diagonalizable), that is,
then,
Writing P as a block matrix of its column vectors
the above equation can be rewritten as
So the column vectors of P are right eigenvectors of A, and the corresponding diagonal entry is the corresponding eigenvalue. The invertibility of P also suggests that the eigenvectors are linearly independent and form the basis of Fn. This is the necessary and sufficient condition for diagonalizability and the canonical approach of diagonalization. The row vectors of P-1 are the left eigenvectors of A.
대각화하는 이유
행렬의 n제곱을 찾기 위해서이다.
대각화하는 방법
우선, 다음 식에서 λ와 x를 찾는다.
이 수식을 다음과 같이 바꿔 쓸 수 있다.
여기서 A-λE의 역행렬이 존재하지 않아야 하므로,
이 된다. 이 수식을 계산하여 λi (i = 1, 2, …, n)를 구한다. (만약 λi가 n개가 나오지 않으면, 대각화할 수 없다.)
그러고 나서, Ax=λx 식에 각각 대입하여 λi에 대응되는 xi (i = 1, 2, …, n)를 구해낸다.
한꺼번에 표현하면 다음과 같다.
References
- http://en.wikipedia.org/wiki/Matrix_diagonalization#Diagonalization
- http://blog.naver.com/lucifer246/166277074