eigenvalues_general_matrixeigenvalues_general_matrixeigenvalues_general_matrixEigenvaluesGeneralMatrixEigenvaluesGeneralMatrix (Operator)

Name

eigenvalues_general_matrixeigenvalues_general_matrixeigenvalues_general_matrixEigenvaluesGeneralMatrixEigenvaluesGeneralMatrix — Compute the eigenvalues and optionally the eigenvectors of a general matrix.

Signature

Description

The operator eigenvalues_general_matrixeigenvalues_general_matrixeigenvalues_general_matrixEigenvaluesGeneralMatrixEigenvaluesGeneralMatrix computes all eigenvalues and, optionally, the left or right eigenvectors of a square, general MatrixMatrixMatrixMatrixmatrix. The matrix is defined by the matrix handle MatrixIDMatrixIDMatrixIDMatrixIDmatrixID. The computed eigenvectors have the norm 1.

The operator generates the new matrices EigenvaluesRealEigenvaluesRealEigenvaluesRealEigenvaluesRealeigenvaluesReal and EigenvaluesImagEigenvaluesImagEigenvaluesImagEigenvaluesImageigenvaluesImag with the real and the imaginary parts of the computed eigenvalues. Each matrix has one column and n rows, where n is the number of rows of the input MatrixMatrixMatrixMatrixmatrix. In contrast to the operator eigenvalues_symmetric_matrixeigenvalues_symmetric_matrixeigenvalues_symmetric_matrixEigenvaluesSymmetricMatrixEigenvaluesSymmetricMatrix, the order of the eigenvalues is not defined. The operator returns the matrix handles EigenvaluesRealIDEigenvaluesRealIDEigenvaluesRealIDEigenvaluesRealIDeigenvaluesRealID and EigenvaluesImagIDEigenvaluesImagIDEigenvaluesImagIDEigenvaluesImagIDeigenvaluesImagID. If desired, the real and imaginary parts of the computed eigenvectors are stored in the new matrices EigenvectorsRealEigenvectorsRealEigenvectorsRealEigenvectorsRealeigenvectorsReal and EigenvectorsImagEigenvectorsImagEigenvectorsImagEigenvectorsImageigenvectorsImag. For this, the operator returns valid matrix handles EigenvectorsRealIDEigenvectorsRealIDEigenvectorsRealIDEigenvectorsRealIDeigenvectorsRealID and EigenvectorsImagIDEigenvectorsImagIDEigenvectorsImagIDEigenvectorsImagIDeigenvectorsImagID. Access to the elements of the matrix is possible e.g. with the operator get_full_matrixget_full_matrixget_full_matrixGetFullMatrixGetFullMatrix.

The computation type of eigenvectors can be selected via the parameter ComputeEigenvectorsComputeEigenvectorsComputeEigenvectorsComputeEigenvectorscomputeEigenvectors. If ComputeEigenvectorsComputeEigenvectorsComputeEigenvectorsComputeEigenvectorscomputeEigenvectors = 'none'"none""none""none""none", no eigenvectors are computed. If 'left'"left""left""left""left" is selected, the left eigenvalues are computed. If 'right'"right""right""right""right" is selected, the right eigenvalues are computed.

Example:

           /  6.0  4.0 -8.0  \
  Matrix = |  5.0  7.0  3.0  |     ComputeEigenvectors = 'right'
           \  4.0 -1.0  4.0  /

                         /  3.3110 \                      /  5.4143  \
  ->   EigenvaluesReal = |  3.3110 |    EigenvaluesImag = | -5.4143  |
                         \ 10.3781 /                      \  0.0     /

                          /  0.6353   0.6353   0.4813  \
       EigenvectorsReal = | -0.4764   0.4764   0.8605  |
                          \ -0.0246  -0.0246   0.1669  /

                          /  0.0      0.0      0.0  \
       EigenvectorsImag = | -0.2485   0.2485   0.0  |
                          \ -0.5542   0.5542   0.0  /

Parallelization

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Multithreading scope: global (may be called from any thread).
• Processed without parallelization.

Parameters

Result

If the parameters are valid, the operator eigenvalues_general_matrixeigenvalues_general_matrixeigenvalues_general_matrixEigenvaluesGeneralMatrixEigenvaluesGeneralMatrix returns the value 2 (H_MSG_TRUE). If necessary, an exception is raised.

Possible Predecessors

create_matrixcreate_matrixcreate_matrixCreateMatrixCreateMatrix

Possible Successors

get_full_matrixget_full_matrixget_full_matrixGetFullMatrixGetFullMatrix, get_value_matrixget_value_matrixget_value_matrixGetValueMatrixGetValueMatrix, get_diagonal_matrixget_diagonal_matrixget_diagonal_matrixGetDiagonalMatrixGetDiagonalMatrix

See also

eigenvalues_symmetric_matrixeigenvalues_symmetric_matrixeigenvalues_symmetric_matrixEigenvaluesSymmetricMatrixEigenvaluesSymmetricMatrix, generalized_eigenvalues_symmetric_matrixgeneralized_eigenvalues_symmetric_matrixgeneralized_eigenvalues_symmetric_matrixGeneralizedEigenvaluesSymmetricMatrixGeneralizedEigenvaluesSymmetricMatrix, generalized_eigenvalues_general_matrixgeneralized_eigenvalues_general_matrixgeneralized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrixGeneralizedEigenvaluesGeneralMatrix

References

David Poole: “Linear Algebra: A Modern Introduction”; Thomson; Belmont; 2006.
Gene H. Golub, Charles F. van Loan: “Matrix Computations”; The Johns Hopkins University Press; Baltimore and London; 1996.

Module

Foundation