Chapter 5: Eigenvalues and Eigenvectors: The Directions a Matrix Cannot Turn

A matrix moves the plane, and Chapter 3 taught you to watch the motion: squares shear into parallelograms, grids tilt, almost every vector is swung to point somewhere new. Almost every vector. For most matrices there are a few special directions that the transformation is powerless to turn — feed in a vector along one of them and the output points along the very same line, merely stretched, shrunk, or flipped. These are the eigenvectors, and the stretch factors attached to them are the eigenvalues. They are the axes of a matrix's private geometry, and finding them turns an opaque grid of entries into a legible motion.

Special directions should be hunted, not announced. This chapter's figures draw a matrix as a field of arrows — at every point, an arrow shows where the matrix sends that position vector — because in a field the invariant lines give themselves away: they are the places where the arrows lie along the line they sit on, instead of crossing it. An eigenvalue is visible too, as arrow length and arrow direction: outward for positive, inward for negative, long for large.

The hunt has three episodes. First you will inspect a $3\times 3$ matrix's field and catch its three invariant lines — one of them carrying flipped arrows. Then you will rotate a single test line through a continuum of candidate angles and discover how rare the invariant ones are, and how the determinant of the last chapter finds them algebraically. Finally you will meet the matrices that have no real invariant direction at all — their eigenvalues are complex, and instead of holding a line fixed they spiral the whole plane.