GATE CS 2014 Set 1 Q.15 (CS Q.5): Linear Algebra

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 _____________________.

Your answer

Official answer and solution

Official answer: 0
From: Official GATE 2014 answer key

Solution

Apply the spectral theorem property that eigenvectors corresponding to distinct eigenvalues of any real symmetric matrix are orthogonal.

Let $A$ be a real symmetric matrix ( $A = A^T$ ).
Let $\lambda_1$ and $\lambda_2$ be two distinct eigenvalues of $A$ ( $\lambda_1 \neq \lambda_2$ ) with corresponding eigenvectors $v_1$ and $v_2$ , so $A v_1 = \lambda_1 v_1$ and $A v_2 = \lambda_2 v_2$ .

Consider the inner product $\langle A v_1, v_2 \rangle$ :

\langle A v_1, v_2 \rangle = (A v_1)^T v_2 = v_1^T A^T v_2 = v_1^T A v_2 = v_1^T (\lambda_2 v_2) = \lambda_2 (v_1^T v_2)

On the other hand:

\langle A v_1, v_2 \rangle = (\lambda_1 v_1)^T v_2 = \lambda_1 (v_1^T v_2)

Equating both expressions:

(\lambda_1 - \lambda_2)(v_1^T v_2) = 0

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

v_1^T v_2 = 0

Thus, the dot product of the eigenvectors is 0.

Final answer

Common trap

Confusing symmetric positive definite matrices with arbitrary matrices where eigenvectors are not necessarily orthogonal.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 1
Question: 15 of 65, printed as CS Q.5
Section: Computer Science and Information Technology

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