GATE ME 2021 Set 1 — Question 36
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Consider a vector in 2-dimensional space. Let its direction (counter-clockwise angle with the positive -axis) be . Let be an eigenvector of a matrix with corresponding eigenvalue . If we denote the magnitude of a vector by , identify the VALID statement regarding , where .
Correct answer
(B) Direction of p' = θ, \ p'\ = λ\ p\
Solution
By the definition of an eigenvector, if is an eigenvector of matrix with eigenvalue , then:Given , we have .
Since , the vector is a positive scalar multiple of . This means points in the same direction as . Therefore, the direction of is the same as the direction of , which is .
The magnitude of is:.
Thus, the direction is and the magnitude is .
Since , the vector is a positive scalar multiple of . This means points in the same direction as . Therefore, the direction of is the same as the direction of , which is .
The magnitude of is:.
Thus, the direction is and the magnitude is .
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