101 lines
4.5 KiB
Markdown
101 lines
4.5 KiB
Markdown
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# EigenVectorsTriangularZ
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Computes eigenvectors of n real upper quasi-triangular or complex upper triangular matrix computed by [EigenHessenbergSchurQ](/en/docs/matrix/openblas/eigen_values/general_matrices/eigenhessenbergschurq) or [EigenSolverSchur](/en/docs/matrix/openblas/eigen_values/general_matrices/eigensolverschur).
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A = Q * T * Q**H, where T is an upper quasi-triangular matrix (the Schur form), and Q is the orthogonal matrix of Schur vectors.
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LAPACK function [TREVC](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/trevc.html).
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Computing for type matrix<double>
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```
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bool matrix::EigenVectorsTriangularZ(
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ENUM_EIG_VECTORS side, // compute left and/or right eigenvectors
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matrix& schur_z, // orthogonal matrix of Schur vectors
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matrix& left_eigenvectors, // matrix of computed left vectors
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matrix& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<float>
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```
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bool matrixf::EigenVectorsTriangularZ(
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ENUM_EIG_VECTORS side, // compute left and/or right eigenvectors
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matrixf& schur_z, // orthogonal matrix of Schur vectors
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matrixf& left_eigenvectors, // matrix of computed left vectors
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matrixf& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<complex>
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```
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bool matrixc::EigenVectorsTriangularZ(
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ENUM_EIG_VECTORS side, // compute left and/or right eigenvectors
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matrixc& schur_z, // orthogonal matrix of Schur vectors
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matrixc& left_eigenvectors, // matrix of computed left vectors
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matrixc& right_eigenvectors // matrix of computed right vectors
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);
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```
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Computing for type matrix<complexf>
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```
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bool matrixcf::EigenVectorsTriangularZ(
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ENUM_EIG_VECTORS side, // compute left and/or right eigenvectors
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matrixcf& schur_z, // orthogonal matrix of Schur vectors
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matrixcf& left_eigenvectors, // matrix of computed left vectors
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matrixcf& right_eigenvectors // matrix of computed right vectors
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);
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```
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Parameters
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side
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[in] ENUM_EIG_VECTORS enumeration value which determines what the eigenvectors are computed left, right or both.
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shur_z
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[in] Orthogonal matrix of Schur vectors computed by [EigenHessenbergSchurQ](/en/docs/matrix/openblas/eigen_values/general_matrices/eigenhessenbergschurq) or [EigenSolverSchur](/en/docs/matrix/openblas/eigen_values/general_matrices/eigensolverschur). Can be of zero size. In this case no backtransformation is produced.
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left_eigenvectors
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[out] Matrix of left eigenvectors.
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right_eigenvectors
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[out] Matrix of right eigenvectors.
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Return Value
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Return true if successful, otherwise false in case of an [error](/en/docs/constants/errorswarnings/errorcodes).
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Note
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The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:
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T*x = w*x, (y**H)*T = w*(y**H)
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where y**H denotes the conjugate transpose of y.
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The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.
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This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the orthogonal factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A.
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ENUM_EIG_VECTORS
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An enumeration that specifies whether to calculate eigenvectors.
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| ID | Description |
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| --- | --- |
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| EIGVECTORS_N | Cannot be used with this method |
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| EIGVECTORS_L | Only left eigenvectors are computed. |
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| EIGVECTORS_R | Only right eigenvectors are computed. |
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| EIGVECTORS_LR | Left and right eigenvectors are computed, eigenvalues are always computed. |
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