MATH221 April 2009
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Question 12 (e)
The sum of two eigenvectors of A is again an eigenvector of A.
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The answer is false.
The motivation for why it should be false is as follows: Let be two eigenvectors of A with respective eigenvalues . Then . So unless , will not be an eigenvector.
The above observation will guide us in our construction of a counterexample. Consider
The eigenvalues of A are 1 and 2 with corresponding eigenvectors and .
However, as discussed above, . So is not an eigenvector of A.
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