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Eigenvalues and eigenvectors [2,1,1;1,2,1;…]

Short answer

Eigenvalues: 1, 1, 3. Eigenvectors: for λ=3, k(1,1,0); for λ=1, any combination of (−1,1,0) and (−1,0,1). They allow diagonalization to compute Aⁿ easily, e.g., Aⁿ = P diag(1,1,3ⁿ) P⁻¹.

Find the eigenvalues and eigenvectors of the matrix [2 1 1; 1 2 1; 0 0 1]. Give an example showing the importance of eigenvectors.

A = [2 1 1; 1 2 1; 0 0 1]

Eigenvalues

Expanding |A − λI| along the third row:

|A − λI| = (1 − λ) · | 2−λ  1  |
                     |  1  2−λ | = (1 − λ)[(2 − λ)² − 1] = (1 − λ)(1 − λ)(3 − λ)

Eigenvalues: λ = 1, 1, 3.

Eigenvectors

λ = 3: A − 3I = [−1 1 1; 1 −1 1; 0 0 −2]. From R₃: z = 0; then x = y. X₁ = (1, 1, 0)ᵀ.

λ = 1: A − I = [1 1 1; 1 1 1; 0 0 0]. The single equation x + y + z = 0 (rank 1) has two independent solutions:

  • X₂ = (−1, 1, 0)ᵀ
  • X₃ = (−1, 0, 1)ᵀ
EigenvalueEigenvectors
3k(1, 1, 0)ᵀ
1 (twice)k₁(−1, 1, 0)ᵀ + k₂(−1, 0, 1)ᵀ

Example showing the importance of eigenvectors

Eigenvectors let us diagonalize a matrix and compute its powers easily. Take P = [X₂ X₃ X₁] = [−1 −1 1; 1 0 1; 0 1 0] and D = diag(1, 1, 3). Then A = PDP⁻¹ and

Aⁿ = P Dⁿ P⁻¹ = P diag(1, 1, 3ⁿ) P⁻¹
   = [(3ⁿ+1)/2  (3ⁿ−1)/2  (3ⁿ−1)/2]
     [(3ⁿ−1)/2  (3ⁿ+1)/2  (3ⁿ−1)/2]
     [   0          0          1    ]

For n = 3 this gives A³ = [14 13 13; 13 14 13; 0 0 1], which agrees with multiplying A three times. Without eigenvectors we would have to multiply the matrix n times. The same idea is used to solve systems of differential equations X′ = AX, to find principal axes of a quadratic form, and to study vibrations and stability.

✓ Answer checked by Claude Opus 5.5 (high effort)

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