The value of the dot product of the eigenvectors corresponding to any pair of…
2014
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is _____________________.
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Correct answer: 0
Answer: 0 (the eigenvectors are orthogonal).
Reason: For a symmetric matrix, eigenvectors corresponding to distinct eigenvalues are orthogonal. A brief derivation:
Let v and w be eigenvectors with eigenvalues λ and μ respectively, with λ ≠ μ.
Compute vᵀAw in two ways. Using Aw = μw gives vᵀAw = μ vᵀw. Using Av = λv and symmetry of A gives vᵀAw = (Av)ᵀw = λ vᵀw.
Thus λ vᵀw = μ vᵀw, so (λ − μ) vᵀw = 0. Since λ ≠ μ, we conclude vᵀw = 0.
Note: Positive definiteness ensures the eigenvalues are positive, but the orthogonality conclusion follows from symmetry alone.