GATE EC 2023 Set 1 — Question 43
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Control Systems → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The state equation of a second order system is , is the initial condition. Suppose and are two distinct eigenvalues of and and are the corresponding eigenvectors. For constants and , the solution, , of the state equation is
Correct answer
(A) Σᵢ₌₁² αᵢ e^(λᵢ t) vᵢ
Solution
The solution to the linear homogeneous state equation with initial condition is given by:Given that has distinct eigenvalues and corresponding eigenvectors , any initial state can be uniquely represented as a linear combination of these eigenvectors:Substituting this into the solution expression:By the definition of eigenvalues and eigenvectors, . Therefore:This matches option (A).
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