# EigenVectorsTriangularZ

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).

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.

LAPACK function [TREVC](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/trevc.html).

Computing for type matrix<double>

```
bool  matrix::EigenVectorsTriangularZ(
   ENUM_EIG_VECTORS      side,                    // compute left and/or right eigenvectors
   matrix&               schur_z,                 // orthogonal matrix of Schur vectors
   matrix&               left_eigenvectors,       // matrix of computed left vectors
   matrix&               right_eigenvectors       // matrix of computed right vectors
   );

```

Computing for type matrix<float>

```
bool  matrixf::EigenVectorsTriangularZ(
   ENUM_EIG_VECTORS      side,                    // compute left and/or right eigenvectors
   matrixf&              schur_z,                 // orthogonal matrix of Schur vectors
   matrixf&              left_eigenvectors,       // matrix of computed left vectors
   matrixf&              right_eigenvectors       // matrix of computed right vectors
   );

```

Computing for type matrix<complex>

```
bool  matrixc::EigenVectorsTriangularZ(
   ENUM_EIG_VECTORS      side,                    // compute left and/or right eigenvectors
   matrixc&              schur_z,                 // orthogonal matrix of Schur vectors
   matrixc&              left_eigenvectors,       // matrix of computed left vectors
   matrixc&              right_eigenvectors       // matrix of computed right vectors
   );

```

Computing for type matrix<complexf>

```
bool  matrixcf::EigenVectorsTriangularZ(
   ENUM_EIG_VECTORS      side,                    // compute left and/or right eigenvectors
   matrixcf&             schur_z,                 // orthogonal matrix of Schur vectors
   matrixcf&             left_eigenvectors,       // matrix of computed left vectors
   matrixcf&             right_eigenvectors       // matrix of computed right vectors
   );

```

Parameters

side

[in]  ENUM_EIG_VECTORS enumeration value which determines what the eigenvectors are computed left, right or both.

shur_z

[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.

left_eigenvectors

[out] Matrix of left eigenvectors.

right_eigenvectors

[out] Matrix of right eigenvectors.

Return Value

Return true if successful, otherwise false in case of an [error](/en/docs/constants/errorswarnings/errorcodes).

Note

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x,     (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of y.

The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.

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.

ENUM_EIG_VECTORS

An enumeration that specifies whether to calculate eigenvectors.

| ID | Description |
| --- | --- |
| EIGVECTORS_N | Cannot be used with this method |
| EIGVECTORS_L | Only left eigenvectors are computed. |
| EIGVECTORS_R | Only right eigenvectors are computed. |
| EIGVECTORS_LR | Left and right eigenvectors are computed, eigenvalues are always computed. |