Eigenvalues and Eigenvectors

• An eigenvector is a non-zero vector whose direction is unchanged by a transformation — its image is just a scaled copy of itself.
• The scale factor is the eigenvalue, λ. The equation Ae = λe defines this relationship.
• Eigenvalues are found by solving the characteristic equation det(A − λI) = 0.
• For a 2×2 matrix this gives a quadratic in λ; for a 3×3 matrix, a cubic.
• Orthogonal matrices are built from mutually perpendicular unit vectors — their inverse equals their transpose.
• Eigenvectors of a symmetric matrix (belonging to different eigenvalues) are always orthogonal — this is the key theorem of the chapter.
• Diagonalising A means finding an orthogonal matrix P (columns = normalised eigenvectors) so that P^TAP = D, a diagonal matrix of eigenvalues.
• The whole method scales up from 2×2 to 3×3 matrices with no new ideas — just more algebra.

1. Eigenvectors — the "unmoved directions"

Picture a transformation T stretching, squashing and rotating the whole plane (or space). Most vectors get sent off in some completely new direction. But for a handful of special directions, T doesn't rotate the vector at all — it just makes it longer, shorter, or flips it. That special vector e is an eigenvector of T.

If e is an eigenvector, its image e′ must be parallel to e — so e′ is just a multiple of e:

Here λ (lambda) is the eigenvalue — the scale factor along that direction. Geometrically, e defines a whole line through the origin, and that entire line maps onto itself (it's called an invariant line). If λ = 1, every individual point on that line stays exactly where it is — a line of invariant points.

2. The characteristic equation

Starting from Ae = λe, rearrange to get everything on one side:

Since eigenvectors are defined to be non-zero, this equation must have a non-zero solution for e. A matrix equation Mx = 0 only has non-zero solutions for x when M is singular (det(M) = 0). So:

Solving this equation (a quadratic for a 2×2 matrix, a cubic for a 3×3 matrix) gives the eigenvalues. Each eigenvalue is then substituted back into (A − λI)e = 0 to find its corresponding eigenvector.