Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3\times 3$ real symmetric matrix. Which one of the following statements is true?

Symmetric Matrix Eigenvectors Property

A fundamental property of real symmetric matrices states that eigenvectors corresponding to distinct eigenvalues are orthogonal.

Eigenvector Orthogonality Condition

Let A be a real symmetric matrix. If $v_1$ and $v_2$ are eigenvectors associated with different eigenvalues (${\lambda }_1\ne {\lambda }_2$), then they are orthogonal. Orthogonality between two vectors $v_1$ and $v_2$ means their scalar product (or dot product) is zero.

This condition is mathematically expressed as:

$v_1^T{v}_2=0$

Applying the Property to the Question

The question involves a $3\times 3$ real symmetric matrix and its eigenvectors $v_1$ and $v_2$, which correspond to distinct eigenvalues.

Given these conditions, the eigenvectors $v_1$ and $v_2$ must satisfy the orthogonality property.

Result

Therefore, the correct statement is that the dot product of the eigenvectors is zero:

$v_1^T{v}_2=0$