A real 2×2 matrix with one repeated eigenvalue can behave in two fundamentally different ways. A uniform-scaling matrix has every nonzero vector as an eigenvector, while a defective matrix has only one eigendirection. The defective case leads naturally to a Jordan block, a generalized eigenvector and the similarity factorization A = CJC⁻¹.
For a real 2×2 matrix A = [[a,b],[c,d]], a repeated eigenvalue occurs when (a − d)² + 4bc = 0.
The same framework explains both how to construct a basis for a defective matrix and why its powers and exponentials have a simple canonical form.
Let B = A − λI for a defective 2×2 matrix with repeated eigenvalue λ. Since B² = 0 but B ≠ 0, choose w with Bw ≠ 0 and set v = Bw. Then Bv = 0, so v is an eigenvector. Because Bw ≠ 0, w cannot be a scalar multiple of v; therefore C = [v | w] is invertible and gives A = CJC⁻¹.
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Once A = CJC⁻¹, powers reduce to powers of the Jordan block: Aⁿ = CJⁿC⁻¹. For n ≥ 2, Jⁿ has λⁿ on the diagonal and nλⁿ⁻¹ in the upper-right entry. More generally, when λ ≠ 0 and λᵗ is defined, the same pattern gives the corresponding real or complex power Jᵗ and hence Aᵗ = CJᵗC⁻¹.