Say, we have a linear transformation $T:V\to V$ and $A$ is the matrix of $T$ w.r.t. standard basis for $V$. Do eigenvectors $v_1$ and $v_2$ are eigenvectors for $T$ associated to eigenvalues $\lambda_1$ and $\lambda_2$ then would $v_1 + v_2$ be an eigenvector for $T$ with associated eigenvalue $\lambda_1 + \lambda_2$ ?
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$\begingroup$ Go back to the basic definition of an eigenvector and try it. $\endgroup$ – amd Nov 3 '16 at 2:28
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$\begingroup$ Try the identity matrix... $\endgroup$ – user251257 Nov 3 '16 at 2:43
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$\begingroup$ @SFL Do you mean that this is a case where $\lambda_1,\lambda_2$ and $ \lambda_1+ \lambda_2$ are all eigenvalues for $A$ at the same time ? $\endgroup$ – Widawensen Nov 3 '16 at 9:57
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$\begingroup$ @Widawensen I mean that if they are both eigenvalues is the addition also an eigenvalue $\endgroup$ – SFL Nov 3 '16 at 11:07
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$\begingroup$ @SFL addition not always is an eigenvalue, If it would be a case we would have an infinite number of eigenvalues for any matrix ( all possible additions) $\endgroup$ – Widawensen Nov 3 '16 at 12:24
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Not necessarily: if $T v_1 = \lambda_1 v_1$ and $T v_2 = \lambda_2 v_2$ then
$$ T(v_1 + v_2) = \lambda_1 v_1 + \lambda_2 v_2 \ne (\lambda_1 + \lambda_2)(v_1 + v_2) $$
