When a matrix has no repeated eigenvalues, the eigenvectors are always independent and the eigenvector matrix V diagonalizes the original matrix A if applied as a similarity transformation. If B is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem Because B can be singular, an alternative algorithm, called the QZ method, is necessary. The values of that satisfy the equation are the generalized eigenvalues and the corresponding values of x are the generalized right eigenvectors. The generalized eigenvalue problem is to determine the nontrivial solutions of the equation where both A and B are n-by- n matrices and is a scalar. In MATLAB, the function eig solves for the eigenvalues, and optionally the eigenvectors x. The n values of that satisfy the equation are the eigenvalues, and the corresponding values of x are the right eigenvectors. Where A is an n-by- n matrix, x is a length n column vector, and is a scalar. Remarks The eigenvalue problem is to determine the nontrivial solutions of the equation: The eigenvectors are scaled so that the norm of each is 1.0. Produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A* V = B* V* D. Returns a vector containing the generalized eigenvalues, if A and B are square matrices. See the balance function for more details. However, if a matrix contains small elements that are really due to roundoff error, balancing may scale them up to make them as significant as the other elements of the original matrix, leading to incorrect eigenvectors. Ordinarily, balancing improves the conditioning of the input matrix, enabling more accurate computation of the eigenvectors and eigenvalues. Use = eig(A') W = W' to compute the left eigenvectors, which satisfyįinds eigenvalues and eigenvectors without a preliminary balancing step. Matrix V is the modal matrix-its columns are the eigenvectors of A. Matrix D is the canonical form of A-a diagonal matrix with A's eigenvalues on the main diagonal. Produces matrices of eigenvalues ( D) and eigenvectors ( V) of matrix A, so that A* V = V* D. Returns a vector of the eigenvalues of matrix A. Eig (MATLAB Function Reference) MATLAB Function Reference
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