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 ________ .

A real symmetric matrix $A$ has the fundamental property that eigenvectors corresponding to distinct (different) eigenvalues are mutually orthogonal.

Let $\lambda_1 \neq \lambda_2$ be two distinct eigenvalues of a real symmetric matrix $A$, with corresponding eigenvectors $v_1$ and $v_2$:

Taking the transpose and multiplying:

Since $A = A^T$, $v_1^T A v_2 = (A v_1)^T v_2$, which yields:

Since $\lambda_1 \neq \lambda_2$, we must have:

Therefore, the dot product is 0.