GATE EC 2023 Set 1 — Question 15
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let the sets of eigenvalues and eigenvectors of a matrix be and , respectively. For any invertible matrix , the sets of eigenvalues and eigenvectors of the matrix , where , respectively, are
Correct answer
(C) \λₖ 1 ≤ k ≤ n\ and \Pvₖ 1 ≤ k ≤ n\
Solution
We are given that . This means that matrices and are similar matrices.Eigenvalues of similar matrices:
Similar matrices have the same eigenvalues. Therefore, the eigenvalues of are the same as the eigenvalues of , which are .Eigenvectors of similar matrices:
Let be an eigenvalue of with corresponding eigenvector . So, .
We have the relationship . We want to find the eigenvector of corresponding to the eigenvalue .
So, .
Substitute into the equation:
Multiply both sides by from the left:
Comparing this equation with , we can see that must be an eigenvector of corresponding to . Thus, .
To find , multiply by from the left:
So, the eigenvectors of are .Combining both, the sets of eigenvalues and eigenvectors of are and , respectively.
Similar matrices have the same eigenvalues. Therefore, the eigenvalues of are the same as the eigenvalues of , which are .Eigenvectors of similar matrices:
Let be an eigenvalue of with corresponding eigenvector . So, .
We have the relationship . We want to find the eigenvector of corresponding to the eigenvalue .
So, .
Substitute into the equation:
Multiply both sides by from the left:
Comparing this equation with , we can see that must be an eigenvector of corresponding to . Thus, .
To find , multiply by from the left:
So, the eigenvectors of are .Combining both, the sets of eigenvalues and eigenvectors of are and , respectively.
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