Modal matrix of a matrix [1,0,−1;1,2,1;…]

The modal matrix P of A is the matrix whose columns are the linearly independent eigenvectors of A; then P⁻¹AP = D, the diagonal matrix of eigenvalues.

A = [1 0 −1; 1 2 1; 2 2 3]

Eigenvalues

The characteristic equation is λ³ − 6λ² + 11λ − 6 = 0 (trace 6, sum of principal minors 11, |A| = 6), i.e. (λ − 1)(λ − 2)(λ − 3) = 0.

λ = 1, 2, 3.

Eigenvectors, from (A − λI)X = 0

λ = 1: A − I = [0 0 −1; 1 1 1; 2 2 2]. From R₁: z = 0; from R₂: x + y = 0. X₁ = (1, −1, 0)ᵀ.

λ = 2: A − 2I = [−1 0 −1; 1 0 1; 2 2 1]. x + z = 0 and 2x + 2y + z = 0. Take z = 2: x = −2, y = 1. X₂ = (−2, 1, 2)ᵀ.

λ = 3: A − 3I = [−2 0 −1; 1 −1 1; 2 2 0]. z = −2x and x + y = 0. Take x = −1: y = 1, z = 2. X₃ = (−1, 1, 2)ᵀ.

Modal matrix

P = [ 1 −2 −1]
    [−1  1  1]
    [ 0  2  2]      |P| = −2 ≠ 0

Since |P| ≠ 0 the eigenvectors are independent, and

P⁻¹AP = D = [1 0 0]
            [0 2 0]
            [0 0 3]

Answer: the modal matrix is P above (columns X₁, X₂, X₃), and the diagonal form is diag(1, 2, 3). Any non-zero scalar multiples of the columns are also valid.