Tham khảo tài liệu 'ogata - modern control engineering part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | then the statement IX D eig A gives the following result X D eig A X - - - - - D - 0 0 0 - 0 0 0 - The eigenvectors are scaled so that the norm of each is If the eigenvalues of a matrix are distinct the eigenvectors are always independent and the eigenvector matrix X will diagonalize the original matrix A if applied as a similarity transformation matrix. However if a matrix has repeated eigenvalues it is not diagonalizable unless it has a full independent set of eigenvectors. If the eigenvectors are not independent the original matrix is said to be defective. Even if a matrix is defective the solution from eig satisfies the relationship AX XD. Generalized eigenvalues and generalized eigenvectors. If A and B are square matrices then the command eig A B returns a vector containing the generalized eigenvalues solving the equation Ax ABx where X is a scalar. The values of X that satisfy the equation are the generalized eigenvalues and the corresponding values of X are the generalized eigenvectors. To obtain eigenvectors use the double-assignment command as follows X D eig A B This produces a diagonal matrix D of generalized eigenvalues and a square matrix X whose columns are corresponding eigenvectors so that AX BXD Section A-3 Computing Matrix Functions 969 For example if A B - õ ĩ Õ 0 0 1 -4 -6 -4 1 0 1 0 1 0 0 0 1 then eig A B gives eig A B ans - - 0 6258 - and X D eig A B gives X D eig A B X - 78 - - - - - 0 0776 - - - - - D - 0 0 0 0 0 0 - - The eigenvectors are scaled so that the norm of each is 1. Characteristic equation. The roots of the characteristic equation are the same as the eigenvalues of matrix A. The characteristic equation of matrix A is
