Modal matrix of a matrix [5,4;1,2]

The modal matrix P has the eigenvectors of A as its columns, and P⁻¹AP = D is diagonal.

A = [5 4; 1 2]

Eigenvalues

|A − λI| = (5 − λ)(2 − λ) − 4 = λ² − 7λ + 6 = (λ − 1)(λ − 6) = 0

λ = 1, 6.

Eigenvectors

λ = 1: (A − I)X = [4 4; 1 1]X = 0 ⇒ x + y = 0. X₁ = (1, −1)ᵀ.

λ = 6: (A − 6I)X = [−1 4; 1 −4]X = 0 ⇒ x = 4y. X₂ = (4, 1)ᵀ.

Modal matrix and diagonal matrix

P = [ 1  4]        |P| = 1 + 4 = 5 ≠ 0
    [−1  1]

P⁻¹ = (1/5) [1 −4]
            [1  1]

Verification:

AP = [5−4   20+4]  = [1 24]
     [1−2    4+2]    [−1 6]

P⁻¹AP = (1/5)[1 −4; 1 1][1 24; −1 6] = (1/5)[5 0; 0 30] = [1 0; 0 6]

Answer: P = [1 4; −1 1] and D = P⁻¹AP = diag(1, 6).