Directions that only stretch
Picture a transformation acting on the plane. Almost every arrow gets rotated off its original line. But for special directions, the arrow stays on its own line and merely grows, shrinks, or flips. Such a direction is an eigenvector, and the number telling how much it scaled is its eigenvalue, written lambda.
A * v = lambda * v the matrix A acting on v gives the SAME direction v, just scaled by the number lambda
Finding them with the characteristic equation
Rewrite A*v = lambda*v as (A - lambda*I)*v = 0. For a nonzero v to be crushed to zero, the matrix (A - lambda*I) must collapse space — meaning its determinant is zero. Setting det(A - lambda*I) = 0 gives the characteristic polynomial, whose roots are the eigenvalues.
- Start with A = [[2,1],[0,3]] and form A - lambda*I = [[2-lambda,1],[0,3-lambda]].
- Take the determinant: (2-lambda)(3-lambda) - 1*0 = (2-lambda)(3-lambda).
- Set it to 0; the roots are lambda = 2 and lambda = 3 — the two eigenvalues.
- For lambda = 2 solve (A - 2I)*v = 0 to get the eigenvector (1,0); for lambda = 3 you get (1,1).
An honest warning: not every matrix has real eigenvectors
Eigenvectors are directions that stay on their own line. A pure rotation moves every direction off its line — that is the whole point of rotating. So over the real numbers a rotation has no eigenvectors at all, and its characteristic polynomial has no real roots.
rotation by 90 degrees: A = [[0,-1],[1,0]] det(A - lambda*I) = lambda^2 + 1 = 0 no real solution (roots are imaginary) => no real eigenvector