GATE Electrical 2025 | Question: 2

Question

Let $v_{1}$ and $v_{2}$ be the two eigenvectors corresponding to distinct eigenvalues of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?

• A. $v_{1}^{T} v_{2} \neq 0$
• B. $v_{1}^{T} v_{2}=0$
• C. $v_{1}+v_{2}=0$
• D. $v_{1}-v_{2}=0$

Answer 1

Orthogonality of Eigenvectors: For any real symmetric matrix, eigenvectors corresponding to distinct (different) eigenvalues are always orthogonal to each other. Mathematically, if $v_1$ and $v_2$ are such eigenvectors, their dot product is zero: $v_1 \cdot v_2 = 0$ or $v_1^T v_2 = 0$.

option $B$ is correct.

Answer 2

$Av_1 = a v_1,; Av_2 = b v_2$, with $A^T = A$.

 

Then $v_1^T A = a v_1^T$.

 

So $v_1^T A v_2 = a, v_1^T v_2$ and also $v_1^T A v_2 = v_1^T (b v_2) = bv_1^T v_2$.

 

Hence $(a-b) v_1^T v_2 = 0$. Since $a \ne b$, we get $v_1^T v_2 = 0$. Option B