Geometry and algebra of linear maps, vectors, and matrices
Most vectors get knocked off course when a matrix acts on them: they rotate as well as stretch. But a few special directions are invariant. The matrix only stretches or flips them, never turns them. Those are the eigenvectors, and the stretch factor is the eigenvalue.
Read it aloud: applying A to its eigenvector v gives back the same direction, just scaled by λ. If λ = 2, that direction doubles; if λ = −1, it flips; if λ = 0.5, it shrinks by half. The eigenvectors form the skeleton of the transformation, the axes along which it acts most simply.
Drag a vector around the figure. Most directions visibly rotate under A; only along the eigenvector directions does the output stay parallel to the input.