Calculus Diagonalization

Question

M = [11 2] Find formulas for the entries of Mⁿ, where n is a positive integer.

[-4 5]

Answer 1

As the title of the questions suggests, we need to start with the diagonalization of the matrix. Diagonalization means to express the matrix in the following form: M = VΛV^-1, where V = [v₁ v₂] (v₁,v₂ are eigenvectors) and Λ = diag(λ₁, λ₂). Therefore, we need to find the eigenvalues and eigenvectors of the matrix. By setting det(M - λI) to 0, we get λ = 7, 9. Plugging these eigenvalues into the definition of eigenvalues and eigenvectors, Av = λv, we get v₁=[-1 1]^T and v₂=[-1 2]^T. By multiplying by V and V^-1 of both sides of the equation M = VΛV^-1, we get V^-1MV = Λ. Taking both sides to the power of n, on the LHS we have V^-1MV * V^-1MV ... V^-1MV which simplifies to

V^-1MⁿV. On the RHS we have, Λⁿ = diag(λ₁ⁿ, λ₂ⁿ). Multiplying again both sides by V and V^-1 we get, Mⁿ = VΛⁿV^-1. Carrying out the matrix multiplication we get

Mⁿ = [-7ⁿ + 2 * 9ⁿ, -7 + 9ⁿ]

[2(7ⁿ-9ⁿ), 2 * 7ⁿ - 9ⁿ]