A real 3×3 matrix has a cubic characteristic polynomial, and every real cubic has at least one real root — so however you stretch, shear and spin space, some line through the origin always survives, its points only scaled. What becomes of the other two directions is the interesting part: either they are two more axes, or they pair off into a plane the map spins within but never leaves. Set A, then watch drifting directions fall onto whichever one wins.
eigen-axes — Av = λv, dot sits at λvinvariant plane — the complex pair's spiralv — the probe you aimAv — its image (green when aligned)A · sphere — ellipsoid of singular valuesdrift — directions flowing to the dominant subspaceEigen-directions solve det(A − λI) = 0, a cubic — and a real cubic always has a real root, so every real 3×3 map fixes at least one line. The other two roots are either real as well, or a conjugate pair a ± bi with no real eigenvector at all; what such a pair leaves invariant is a plane, the null space of the real matrix A² − 2aA + (a²+b²)I. Restricted to that plane A is a rotation by θ = atan2(b, a) composed with a scale by r = √(a²+b²) — a logarithmic spiral, on which nothing holds still yet nothing escapes. When a repeated λ has an eigenspace smaller than its multiplicity the matrix is defective (the Shear preset): no eigenbasis exists, and the Jordan and real Schur forms are what you reach for instead. Note the eigen-axes are not the ellipsoid's axes — that ellipsoid is the image of the unit sphere, so its axes are the singular values of A. The two agree only when A is symmetric, and that agreement is the spectral theorem: symmetric maps get a full orthogonal eigenbasis. It's why an inertia tensor has principal axes and a covariance matrix has principal components — and why ρ(A) = max|λ|, alone, decides whether Ak decays to nothing or runs away.