Computing A^100 directly means ninety-nine matrix multiplications. Through A = PDP⁻¹ it takes one hundred scalar powers and two multiplications. Why does the change of basis make the work collapse?

D. Because in an eigenvector basis the matrix only scales each axis.

A² = PDP⁻¹PDP⁻¹ = PD²P⁻¹, each interior P⁻¹P becomes the identity and disappears, and D^100 is nearly free, since a diagonal matrix has no interaction between entries. Underneath, the matrix stretches each eigen-axis by its own factor and P is only a translation into that coordinate system. The same idea decouples systems of differential equations.