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Updated SelfAdjointEigenSolver documentation to include that the eigenvectors matrix is unitary.

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Theo Fletcher 2021-03-16 03:12:42 +00:00 committed by TheoFletcher
parent 2e83cbbba9
commit b8502a9dd6
1 changed files with 11 additions and 2 deletions
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13
Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h
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@ -44,10 +44,14 @@ ComputationInfo computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag
* \f$ v \f$ such that \f$ Av = \lambda v \f$. The eigenvalues of a * \f$ 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 * 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 * 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 * eigenvectors as its columns, then \f$ A = V D V^{-1} \f$. This is called the
* matrices, the matrix \f$ V \f$ is always invertible). This is called the
* eigendecomposition. * 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$.
*
* The algorithm exploits the fact that the matrix is selfadjoint, making it * The algorithm exploits the fact that the matrix is selfadjoint, making it
* faster and more accurate than the general purpose eigenvalue algorithms * faster and more accurate than the general purpose eigenvalue algorithms
* implemented in EigenSolver and ComplexEigenSolver. * implemented in EigenSolver and ComplexEigenSolver.
@ -256,6 +260,11 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
* matrix \f$ A \f$, then the matrix returned by this function is the * matrix \f$ 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$. * matrix \f$ 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 * Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
* Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out * Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
* *