2.2.5—Eigenvalues and eigenvectors

If A has a basis of independent eigenvectors, place them as columns of P and their eigenvalues in matching diagonal positions of D. Then A=PDP⁻¹ and A^n=PD^nP⁻¹.

The order of columns in P must match the order of eigenvalues in D. Diagonal powers are easy, which is why diagonalisation helps with recurrences and repeated transformations.

If P=[v₁ v₂] and Av₁=3v₁, Av₂=−v₂, then D=diag(3,−1); A^n acts by multiplying the two eigen-components by 3^n and (−1)^n.

A matrix with eigenvalues is not automatically diagonalizable; it needs enough linearly independent eigenvectors.