GATE EE 2016 Set 2 — Question 42
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let . Consider the set of all vectors such that where . Then is
Correct answer
(D) an ellipse with minor axis along pmatrix 1 \ 1 pmatrix
Solution
Given , the condition can be written as:Since is symmetric, . Thus,The equation of the set is . This is an ellipse. To find its axes, we find the eigenvalues of . The eigenvalues of are found from . The eigenvalues of are and .The eigenvector for (from with ) is found by , which is .
In the canonical form of an ellipse , the semi-axis length is . The minor axis corresponds to the largest eigenvalue. Since is the largest eigenvalue, the minor axis is along its corresponding eigenvector .
In the canonical form of an ellipse , the semi-axis length is . The minor axis corresponds to the largest eigenvalue. Since is the largest eigenvalue, the minor axis is along its corresponding eigenvector .
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