GATE CS 2014 Set 1 Q5 — Eigenvalues and Eigenvectors, LU Decomposition

For any real symmetric matrix, eigenvectors corresponding to distinct (different) eigenvalues are always orthogonal to each other.

Let $A$ be a $4 \times 4$ symmetric positive definite matrix. Since $A$ is symmetric, we have $A^T = A$.

Let $v_1$ and $v_2$ be two eigenvectors corresponding to distinct eigenvalues $\lambda_1$ and $\lambda_2$ respectively (where $\lambda_1 \neq \lambda_2$):

Taking the transpose on both sides of the first equation:

Since $A$ is symmetric ($A^T = A$):

Now, post-multiplying both sides by $v_2$:

Substituting $A v_2 = \lambda_2 v_2$ into the left-hand side:

Since the eigenvalues are distinct, $\lambda_1 - \lambda_2 \neq 0$. Therefore, we must have:

The dot product of the two eigenvectors is $v_1 \cdot v_2 = v_1^T v_2 = 0$.