The stubborn vectors, eigenvectors and eigenvalues

What you’ll learn

This is lesson 14 of Track 4 (Visual Math: Linear Algebra), and the deepest idea in it. When a transformation moves the plane, most vectors are knocked off their own line, but a few stubborn ones stay on their line and merely get scaled. Those are eigenvectors, and their scale factors are eigenvalues. By the end you will be able to identify a 2x2 matrix’s eigenvectors and read its eigenvalues, both by eye for simple cases and by the characteristic equation det(M - λI) = 0 for the rest, and then diagonalize the matrix in its eigenvector basis, delivering the change-of-basis lesson’s promise that the right basis makes a transformation as simple as possible. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.

Where this fits

This is lesson 14 of 15, the fifth lesson of Phase 3 (advanced perspectives), and the conceptual peak of the track. It sits at the intersection of two earlier lessons: the characteristic equation is the inverses lesson’s collapse condition (det = 0), and diagonalization is the change-of-basis sandwich applied to the basis the transformation prefers. The final lesson, Abstract vector spaces, steps back from arrows entirely and asks which of these ideas survive when a vector is a function or a polynomial.

Before you start

Prerequisites: Change of basis (diagonalization is the M^-1 · A · M sandwich, here written P^-1 · M · P) and Inverses, column space, and null space (the characteristic equation rests on “a nonzero vector maps to zero only when the determinant is zero”, and eigenvectors are found as a null space). You should be comfortable computing a 2x2 determinant and inverting a 2x2 matrix. The practice is pen and paper.

About the math

This lesson does the most genuine computation in the track: solving a characteristic equation (a small quadratic in λ), finding null spaces, and carrying out a diagonalization. The arithmetic stays at the 2x2 level throughout, and the worked examples are chosen so the numbers come out clean. The reward for the effort is the payoff, watching a messy matrix turn into a clean diagonal once you view it in its own eigenvector basis.

By the end, you’ll be able to

• Define an eigenvector and eigenvalue by M·v = λ·v and describe them geometrically
• Spot eigenvectors of simple matrices (stretches, shears, rotations) by eye
• Find eigenvalues by solving the characteristic equation det(M - λI) = 0
• Find each eigenvector as the null space of (M - λI) and verify it
• Diagonalize a matrix with two independent eigenvectors via D = P^-1 · M · P, and connect eigen-analysis to PCA and gradient stability

Time and difficulty

• Read time: about 13 minutes
• Practice time: about 15 minutes (a find-the-eigenvalues-and-eigenvectors exercise, a spot-by-eye and diagonalize drill, and flashcards)
• Difficulty: standard, on the harder end (the track’s conceptual peak and most computation-heavy lesson, though all arithmetic stays at the 2x2 level)