NonsymmetricEigenvalueDecomposition Class

The eigenvalue decomposition of a real square general matrix A rewrites the matrix in the form A = XLX^-1, where X is a matrix and L is a diagonal matrix. The diagonal elements of L are the eigenvalues. The columns of X are the corresponding eigenvectors.

The eigenvalues are either real or come in complex conjugate pairs. The eigenvectors corresponding to real eigenvalues are also real. The eigenvectors corresponding to pairs of complex conjugate eigenvalues are themselves complex conjugates.

The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are orthogonal. Its eigenvalue decomposition can be calculated more easily. This is implemented by the SymmetricEigenvalueDecomposition class.

EigenvalueDecomposition inherits from LinearOperator. However, unlike the other decompositions, the nonsymmetric eigenvalue decomposition does not offer a speed advantage when performing calculations most calculations. For example, solving an equation using the eigen decomposition itself requires the solution of a system of equations. For this reason, the Solve(DenseMatrix, Boolean), GetInverse(Boolean) and EstimateConditionNumber()()()() methods simply call the corresponding method on the base matrix.