Calculate the eigenvalues of a square matrix used in dimensionality reduction.
Eigenvalues of a 2×2 matrix are found by solving the characteristic equation det(A − λI) = 0, which reduces to λ² − trace×λ + det = 0. Using the quadratic formula: λ = (trace ± √(trace² − 4×det)) / 2. Eigenvalues are fundamental in PCA, spectral clustering, and stability analysis of dynamical systems.
Characteristic Equation
lambda = (trace ± sqrt(trace^2 - 4*det)) / 2
In PCA, eigenvalues of the covariance matrix represent the variance explained by each principal component. The largest eigenvalues correspond to the directions of maximum variance in the data.
A negative discriminant means the eigenvalues are complex conjugates. This occurs in matrices representing rotations or oscillatory systems and indicates no real principal directions exist.