GATE EE 2016 Set 2 — Question 59
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Control Systems → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Consider a linear time invariant system , with initial condition at . Suppose and are eigenvectors of matrix corresponding to distinct eigenvalues and respectively. Then the response of the system due to initial condition is
Correct answer
(A) e^(λ₁ t) α
Solution
For a linear time-invariant system , the solution is given by .
If the initial condition is an eigenvector of corresponding to the eigenvalue , then by definition .
Using the power series expansion of the matrix exponential:
Substituting :
.
Thus, the response is .
If the initial condition is an eigenvector of corresponding to the eigenvalue , then by definition .
Using the power series expansion of the matrix exponential:
Substituting :
.
Thus, the response is .
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