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2, April 1973.The following MATLAB function produces the Eigenvalues and Eigenvectors of matrix X. Stewart, "An Algorithm for Generalized Matrix Eigenvalue Problems", SIAM J. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, Lecture Notes in Computer Science, Vol. 6, second edition, Springer-Verlag, 1976. Moler, Matrix Eigensystem Routines - EISPACK Guide, Lecture Notes in Computer Science, Vol. Qz QZ factorization for generalized eigenvalues See Also balance Improve accuracy of computed eigenvaluesĬondeig Condition number with respect to eigenvalues For detailed descriptions of these algorithms, see the EISPACK Guide.ĭiagnostics If the limit of 30 n iterations is exhausted while seeking an eigenvalue: Modifications to the QZ routines handle the special case B = I. When eig is used with one complex argument, the solution is computed using the QZ algorithm as eig(X,eye(X)). When eig is used with two input arguments, the EISPACK routines QZHES, QZIT, QZVAL, and QZVEC solve for the generalized eigenvalues via the QZ algorithm.
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The EISPACK subroutine HQR2 is modified to make computation of eigenvectors optional. HQR2 finds the eigenvalues and eigenvectors of a real upper Hessenberg matrix by the QR method. ORTRAN accumulates the transformations used by ORTHES. ORTHES converts a real general matrix to Hessenberg form using orthogonal similarity transformations. BALANC and BALBAK balance the input matrix. Try the statementsĪlgorithm For real matrices, eig(X) uses the EISPACK routines BALANC, BALBAK, ORTHES, ORTRAN, and HQR2. It is an example for which the nobalance option is necessary to compute the eigenvectors correctly.
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Even if a matrix is defective, the solution from eig satisfies A* X = X* D.Įxamples The matrix B = has elements on the order of roundoff error. If the eigenvectors are not independent then the original matrix is said to be defective.
Eig matlab full#
However, if a matrix has repeated eigenvalues, it is not similar to a diagonal matrix unless it has a full (independent) set of eigenvectors. 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