Definition
A complex matrix is self adjoint if $A = \bar{A}^T$, reading “equal to its own conjugate transpose”. It’s also called “Hermitian”.
For real values, this means $A = A^T$
In Eigen, if you know $A = A^T$, you can use Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> es(cov); to calculate its Eigen Vectors and Eigen Values.
- Reduces A to tridiagonal form (cheaper than the full Hessenberg Reduction that general matrices require)
- Apply Symmetric QR or divide-and-conquer algorithm
It’s roughly twice as fast on symmetric/Hermitian than a general solver.
An Eigen Solver uses an upper Hessenberg form, then runs the QR algorithm
PSD (Positive-Semi-Definite) matrix is a special type of self-adjoint matrix:
\[\begin{gather*} \begin{aligned} & A = A^T \\ & x^T A x \ge 0 \end{aligned} \end{gather*}\]Robust Information Matrix
If we want to get the inverse of a self adjoint matrix robustly, first, we have
\[\begin{gather*} \begin{aligned} V = (U \Sigma U^T) \\ \Rightarrow V^{-1} = (U \Sigma U^T)^T = U \Sigma^{-1} U^T \end{aligned} \end{gather*}\]But we also want to floor the inverse values with an absolute floor value:
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Eigen::Matrix3d robustInfo(const Eigen::Matrix3d &cov,
double rel_floor = 1e-2, // floor as % of σ_max
double abs_floor = 1e-4) // or absolute floor
{
// For symmetric PSD matrices Eigen's SelfAdjointEigenSolver is cheaper,
// gives eigenvalues λ and eigenvectors V (cov = V Λ Vᵀ).
Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> es(cov);
const auto &lambdas = es.eigenvalues(); // ascending order
Eigen::Vector3d inv_lambda;
const double λ_max = lambdas.tail<1>()(0);
const double floor_val = std::max(abs_floor, rel_floor * λ_max);
for (int i = 0; i < 3; ++i)
inv_lambda(i) = 1.0 / std::max(floor_val, lambdas(i));
// info = V · diag(inv_lambda) · Vᵀ (guaranteed symmetric PSD)
return es.eigenvectors() * inv_lambda.asDiagonal() *
es.eigenvectors().transpose();
}
