The condition that a matrix can be diagonalized is that the minimum polynomial of the matrix has no multiple roots,

This is proved in the last section of the unit "λ-matrix". Please have a look.

Prove:

σ^3=σ^2+4σ-4σ

=> polynomial: f (λ) = λ 3-λ 2-4λ+4 = (λ 2) (λ1) (λ+2)

Annihilation Polynomials of Matrix Corresponding to σ

Then the smallest polynomial g(λ) of σ can be divisible by f(λ).

Because f(λ) has no multiple roots, g(λ) has no multiple roots.

So this matrix corresponding to σ can be diagonalized, so σ can also be diagonalized.

Certificate of completion