A real 2×2 matrix with complex eigenvalues has no nonzero real eigenvector direction. Instead, one complex eigenvector provides two real vectors, Re(x) and Im(x), that form a basis in which the matrix becomes a rotation-scaling block. This gives the factorization A = XSX⁻¹ and makes powers and exponentials of A easier to understand.
Complex eigenvalues of a real matrix occur in conjugate pairs and encode rotation together with scaling.
The chapter follows these ideas through symbolic derivations, basis-change diagrams and orbit visualizations.
Write x = Re(x) + i Im(x) and λ = α + iβ. Separating the real and imaginary parts of Ax = λx shows that A maps the pair [Re(x) | Im(x)] according to a real rotation-scaling matrix S. When these two real vectors are linearly independent, they form the columns of X and give A = XSX⁻¹.
The PDF page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.
In A = XSX⁻¹, the matrix S is a rotation-scaling block determined by λ = α + iβ. Its modulus |λ| controls contraction or expansion, while θ = atan2(β, α) controls rotation. Therefore Sᵗ rotates by tθ and scales by |λ|ᵗ, and Aᵗ = XSᵗX⁻¹ carries the same behavior back to the original coordinates. The orbit may appear elliptical in the standard basis even though the underlying action in the S basis is circular.