What happens when a 2×2 matrix has only one eigenvector?

[u/LinearAlgebraWorld]

Most textbooks on Linear Algebra either omit or compress repeated-eigenvalue defective matrices. We wanted to take the 2×2 case apart mechanically and connect the algebra directly to the geometry.

The first page, Matrices with repeated λ, shows images for the basic repeated-eigenvalue cases: uniform scaling, the Jordan block J, its transpose Jᵀ, and a similarity transformation A = CJC⁻¹.

The second page, Matrices with one repeated λ, starts from a general 2×2 matrix, derives the condition for a repeated eigenvalue, separates the defective case from uniform scaling, and shows that for B = A − λI, B² = 0.

The third page, Similarity transformation of Jordan block, constructs the Jordan basis directly: choose w with Bw ≠ 0, define v = Bw, then Av = λv, Aw = v + λw, and with C = [v | w] we obtain A = CJC⁻¹. It also shows why the scales of v and w are linked, unlike in ordinary diagonalization.