The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4 x 4 symmetric positive definite matrix is ___________

GATE 2014 · Engineering Mathematics · Eigen Value · medium

Answer: The dot product of eigenvectors corresponding to different eigenvalues is 0.

1. State the spectral theorem for symmetric matrices: Let lambda1 and lambda2 be two distinct eigenvalues of the symmetric positive definite matrix, with eigenvectors v1 and v2 respectively.
2. Prove orthogonality via eigenvalue equations: Since lambda1 != lambda2, we have (lambda1 - lambda2)*(v1 . v2) = 0, and since lambda1 - lambda2 != 0, it follows that v1 . v2 = 0.