diff options
author | Theo Fletcher <theo_fletcher@live.co.uk> | 2021-03-16 03:12:42 +0000 |
---|---|---|
committer | TheoFletcher <theo_fletcher@live.co.uk> | 2021-03-16 18:48:02 +0000 |
commit | b8502a9dd6fbb721d3586f6f0102ac699f55f743 (patch) | |
tree | 8d97f2e1072ae2a6aa2b179f41f7088ec99c8bd4 /Eigen | |
parent | 2e83cbbba98da3a3f645133352a996b3f6daaed0 (diff) |
Updated SelfAdjointEigenSolver documentation to include that the eigenvectors matrix is unitary.
Diffstat (limited to 'Eigen')
-rw-r--r-- | Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h | 13 |
1 files changed, 11 insertions, 2 deletions
diff --git a/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h index 73b7041e6..56a12d56f 100644 --- a/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h +++ b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h @@ -44,10 +44,14 @@ ComputationInfo computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag * \f$ v \f$ such that \f$ Av = \lambda v \f$. The eigenvalues of a * selfadjoint matrix are always real. If \f$ D \f$ is a diagonal matrix with * the eigenvalues on the diagonal, and \f$ V \f$ is a matrix with the - * eigenvectors as its columns, then \f$ A = V D V^{-1} \f$ (for selfadjoint - * matrices, the matrix \f$ V \f$ is always invertible). This is called the + * eigenvectors as its columns, then \f$ A = V D V^{-1} \f$. This is called the * eigendecomposition. * + * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal + * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then + * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is + * equal to its transpose, \f$ V^{-1} = V^T \f$. + * * The algorithm exploits the fact that the matrix is selfadjoint, making it * faster and more accurate than the general purpose eigenvalue algorithms * implemented in EigenSolver and ComplexEigenSolver. @@ -256,6 +260,11 @@ template<typename _MatrixType> class SelfAdjointEigenSolver * matrix \f$ A \f$, then the matrix returned by this function is the * matrix \f$ V \f$ in the eigendecomposition \f$ A = V D V^{-1} \f$. * + * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal + * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then + * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is + * equal to its transpose, \f$ V^{-1} = V^T \f$. + * * Example: \include SelfAdjointEigenSolver_eigenvectors.cpp * Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out * |