eigenvalues_general_matrixEigenvaluesGeneralMatrixEigenvaluesGeneralMatrixeigenvalues_general_matrixT_eigenvalues_general_matrix🔗

Short description🔗

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

Signature🔗

eigenvalues_general_matrix( matrix MatrixID, string ComputeEigenvectors, out matrix EigenvaluesRealID, out matrix EigenvaluesImagID, out matrix EigenvectorsRealID, out matrix EigenvectorsImagID )

Description🔗

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

The operator generates the new matrices EigenvaluesReal and EigenvaluesImag 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 Matrix. In contrast to the operator eigenvalues_symmetric_matrixEigenvaluesSymmetricMatrix, the order of the eigenvalues is not defined. The operator returns the matrix handles EigenvaluesRealIDeigenvaluesRealIDeigenvalues_real_id and EigenvaluesImagIDeigenvaluesImagIDeigenvalues_imag_id. If desired, the real and imaginary parts of the computed eigenvectors are stored in the new matrices EigenvectorsReal and EigenvectorsImag. For this, the operator returns valid matrix handles EigenvectorsRealIDeigenvectorsRealIDeigenvectors_real_id and EigenvectorsImagIDeigenvectorsImagIDeigenvectors_imag_id. Access to the elements of the matrix is possible e.g., with the operator get_full_matrixGetFullMatrix.

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

Example:

ComputeEigenvectorscomputeEigenvectorscompute_eigenvectors = 'right'"right"

\[\begin{eqnarray*} \texttt{Matrix} = \left[ \begin{array}{rrr} 6.0 & 4.0 & -8.0 \\ 5.0 & 7.0 & 3.0 \\ 4.0 & -1.0 & 4.0 \end{array} \right] \end{eqnarray*}\]

\[\begin{eqnarray*} \to \qquad \texttt{EigenvaluesReal} &=& \left[ \begin{array}{r} 3.3110 \\ 3.3110 \\ 10.3781 \end{array} \right]\\ \texttt{EigenvaluesImag} &=& \left[ \begin{array}{r} 5.4143 \\ -5.4143 \\ 0.0 \end{array} \right]\\ \texttt{EigenvectorsReal} &=& \left[ \begin{array}{rrr} 0.6353 & 0.6353 & 0.4813 \\ -0.4764 & -0.4764 & 0.8605 \\ -0.0246 & -0.0246 & 0.1669 \end{array} \right]\\ \texttt{EigenvectorsImag} &=& \left[ \begin{array}{rrr} 0.0 & 0.0 \quad & \quad 0.0 \\ -0.2485 & 0.2485 \quad & \quad 0.0 \\ -0.5542 & 0.5542 \quad & \quad 0.0 \end{array} \right] \end{eqnarray*}\]

Execution information🔗

Execution information

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

Parameters🔗

MatrixIDmatrixIDmatrix_id (input_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle of the input matrix.

ComputeEigenvectorscomputeEigenvectorscompute_eigenvectors (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)

Computation of the eigenvectors.

Default: 'none'"none"
List of values: 'left', 'none', 'right'"left", "none", "right"

EigenvaluesRealIDeigenvaluesRealIDeigenvalues_real_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the real parts of the eigenvalues.

EigenvaluesImagIDeigenvaluesImagIDeigenvalues_imag_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the imaginary parts of the eigenvalues.

EigenvectorsRealIDeigenvectorsRealIDeigenvectors_real_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the real parts of the eigenvectors.

EigenvectorsImagIDeigenvectorsImagIDeigenvectors_imag_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the imaginary parts of the eigenvectors.

Result🔗

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

Combinations with other operators🔗

Combinations

Possible predecessors

create_matrixCreateMatrix

Possible successors

get_full_matrixGetFullMatrix, get_value_matrixGetValueMatrix, get_diagonal_matrixGetDiagonalMatrix

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