generalized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrixGeneralizedEigenvaluesGeneralMatrixgeneralized_eigenvalues_general_matrix🔗

Short description🔗

generalized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrixGeneralizedEigenvaluesGeneralMatrixgeneralized_eigenvalues_general_matrixT_generalized_eigenvalues_general_matrix — Compute the generalized eigenvalues and optionally the generalized eigenvectors of general matrices.

Signature🔗

generalized_eigenvalues_general_matrix( matrix MatrixAID, matrix MatrixBID, string ComputeEigenvectors, out matrix EigenvaluesRealID, out matrix EigenvaluesImagID, out matrix EigenvectorsRealID, out matrix EigenvectorsImagID )

Description🔗

The operator generalized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrix computes all generalized eigenvalues and, optionally, the left or right generalized eigenvectors of the square, general matrices MatrixA and MatrixB. Both matrices must have identical dimensions. The matrices are defined by the matrix handles MatrixAIDmatrixAIDmatrix_aid and MatrixBIDmatrixBIDmatrix_bid. 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 or columns of the input matrices. In contrast to the operator generalized_eigenvalues_symmetric_matrixGeneralizedEigenvaluesSymmetricMatrix, the order of the generalized 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 respective eigenvectors are stored in the new matrices EigenvectorsReal and EigenvectorsImag. Here, the \(j\)th column of the matrices of eigenvectors contains the related eigenvector to the \(j\)th eigenvalue. For this, the operator returns additionally the 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 or get_sub_matrixGetSubMatrix.

The computation type of eigenvectors can be selected via the parameter ComputeEigenvectorscomputeEigenvectorscompute_eigenvectors. If ComputeEigenvectorscomputeEigenvectorscompute_eigenvectors = 'none'"none", no eigenvectors are computed and the operator is faster. For this, the matrix handles EigenvectorsRealIDeigenvectorsRealIDeigenvectors_real_id and EigenvectorsImagIDeigenvectorsImagIDeigenvectors_imag_id are invalid. If 'right'"right" is selected, the right generalized eigenvalues are computed. The formula for the calculation of the result is

\[\begin{eqnarray*} \texttt{MatrixA} \quad \cdot \quad x_{j} \quad = \quad \lambda_{j} \quad \cdot \quad \texttt{MatrixB} \quad \cdot \quad x_{j}, \end{eqnarray*}\]

with \(\lambda_{j}\) representing the \(j\)th (complex) eigenvalue and \(x_{j}\) represents the corresponding (complex) eigenvector.

If 'left'"left" is selected, the left generalized eigenvalues are computed. The formula for the calculation of the result is

\[\begin{eqnarray*} {x^H}_{j} \quad \texttt{MatrixA} \quad \cdot \quad = \quad {x^H}_{j} \cdot \quad \lambda_{j} \quad \cdot \quad \texttt{MatrixB}, \end{eqnarray*}\]

with \({x^H}_{j}\) represents the conjugate-transposed of \(x_{j}\).

Example:

ComputeEigenvectorscomputeEigenvectorscompute_eigenvectors = 'right'"right"

\[\begin{eqnarray*} \texttt{MatrixA} = \left[ \begin{array}{rrr} 6.0 & 4.0 & -8.0 \\ 5.0 & 7.0 & 3.0 \\ 4.0 & -1.0 & 4.0 \end{array} \right] \qquad \texttt{MatrixB} = \left[ \begin{array}{rrr} 3.0 & 1.0 & 2.0 \\ -5.0 & 7.0 & 2.0 \\ 9.0 & 4.0 & 1.0 \end{array} \right] \end{eqnarray*}\]

\[\begin{eqnarray*} \to \qquad \texttt{EigenvaluesReal} &=& \left[ \begin{array}{r} 0.5363 \\ 0.5363 \\ -6.1616 \end{array} \right]\\ \texttt{EigenvaluesImag} &=& \left[ \begin{array}{r} 0.4208 \\ -0.4208 \\ 0.0 \end{array} \right]\\ \texttt{EigenvectorsReal} &=& \left[ \begin{array}{rrr} 0.3500 & 0.3500 & 0.0410 \\ -0.9565 & -0.9565 & 0.3267 \\ -0.2757 & -0.2757 & -1.0000 \end{array} \right]\\ \texttt{EigenvectorsImag} &=& \left[ \begin{array}{rrr} -0.4644 & 0.4644 \quad & \quad 0.0 \\ 0.0435 & -0.0435 \quad & \quad 0.0 \\ -0.1869 & 0.1869 \quad & \quad 0.0 \end{array} \right] \end{eqnarray*}\]

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🔗

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

Matrix handle of the input matrix A.

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

Matrix handle of the input matrix B.

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 generalized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrix 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

generalized_eigenvalues_general_matrixGeneralizedEigenvaluesGeneralMatrixGeneralizedEigenvaluesGeneralMatrixgeneralized_eigenvalues_general_matrixT_generalized_eigenvalues_general_matrix🔗