eigenvectors and eigenvalues explained
Eigenvalues and eigenvectors are often called the final boss of linear algebra. The names sound intimidating, but the core is simple. A matrix is a transformation that rotates and stretches all of space at once, so most vectors get knocked off their direction by it. A few special directions, however, do not turn at all; they only stretch or shrink along their own line. Those unturned directions are the eigenvectors, and how much they stretch is the eigenvalue. That is all there is to it. And these two pull together what we saw in "Matrices Are Transformations," "The Determinant Is Area Scaling," and "A System Is a Transformation Run Backwards."
Start with the fact that most vectors get turned. Fix one transformation and swing an arrow around the circle: the faint arrow is the original and the gold arrow is the transformed one. At most angles the two point different ways. The transformation has pushed the vector off its original line. Below the figure, the angle between the two arrows is shown, and as you turn the handle that angle is almost always open. What pointed up now points at a slant, and what pointed right gets lifted upward; that is the normal case. A transformation scrambles directions.
Keep turning, though, and at certain special angles the gold arrow lands exactly on the same line as the faint one. The direction is unchanged and only the length has changed; that direction is an eigenvector. The two dashed lines in the figure are those directions, and when the handle reaches one, that line thickens and the readout below announces an eigenvector along with its eigenvalue. It is a direction that holds its own line instead of being swept away by the torrent of the transformation. How much it stretches along that line, the multiplier, is the eigenvalue. As in this figure, a 2D transformation usually has two such directions; here one stretches and the other shrinks slightly. However arbitrary a transformation looks, it has axes of its own: directions it leaves alone.
Look at exactly what an eigenvalue does along its direction. Ride along an eigenvector and the transformation is just multiplication by a number. An eigenvalue of 2 doubles lengths in that direction, 0.5 halves them, and 1 leaves them as they are. A negative eigenvalue flips the vector to the opposite side of the same line: the line is kept, only front and back are swapped. Change the matrix with the four sliders and each faint unit eigenvector and its transformed colored arrow lie on the same dashed line, with both eigenvalues shown below. Push the matrix toward a pure rotation and a warning says there are no real eigenvectors, because no direction can stay on its own line. Along its eigenvectors, a transformation that looked complicated shrinks to a single multiplication.
Here is the truly beautiful part. Redraw the grid along the eigenvector directions instead of the usual horizontal and vertical, and the transformation looks completely different. No rotation, no twisting: each axis is simply stretched by its own eigenvalue. In the coordinate system whose axes are the eigenvectors, that complicated matrix becomes a clean "multiply each axis by a number" form, a diagonal matrix. Switch between the "Standard grid" and "Eigen-axis grid" buttons: on the eigen-axis grid the two colored arrows only stretch along their own axes, and the expression below turns into diagonal form. Eigenvectors are the most natural axes for a transformation, and seen along them any transformation shows its simplest face.
So how do we find these directions? An eigenvector v is a vector for which applying A gives the same result as multiplying by λ, so Av = λv. Rearranged, this becomes (A − λI)v = 0. A nonzero vector v being sent to 0 means the transformation (A − λI) has squashed space. As "A System Is a Transformation Run Backwards" showed, its determinant must then be 0. So solving det(A − λI) = 0 for λ gives the eigenvalues. Move λ with the slider: the unit square under that matrix flattens and swells, and the moment λ reaches an eigenvalue it collapses onto a line and "eigenvalue!" appears. The chain we met earlier, determinant 0 = flattened = cannot run backward, was the key to finding eigenvalues. As a bonus, the determinant is the product of all the eigenvalues, so the area scale factor equals the product of the stretch factors along the axes. The check in the bottom line is exactly that; it all connects.