Eigenvalues and Eigenvectors of a Matrix
- For a square matrix A,
vectors satisfying this condition are called the eigenvectors of the matrix A, and the lambda scalar coefficients are called the eigenvalues.
- An
matrix has
different eigenvectors. However, the corresponding lambda eigenvalues for these eigenvectors may not be different.
- State of a system can be expressed in terms of the eigenvectors of the system of linear equations and in terms of their eigenvalues for the measured quantities.
- Create an
matrix by arranging eigenvectors side by side:
- When we multiply the matrix
with the matrix
and its inverse matrix
from both sides, we get
- That is, the similarity transformation with the eigenvectors matrix makes
as being diagonalized and the elements on the diagonal become the eigenvalues of
.
- In principle, the eigenvalue problem is easy to solve. So-called characteristic equation is to be solved:
- After finding the roots of this n-degree polynomial equation, the corresponding eigenvectors can be obtained by solving the following system of equations:
- Since this method requires determinant calculation, it is not useful for large dimensional matrices.
Subsections