# Properties of eigenvalues/eigenvectors

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 $\lambda_1 + \lambda_2$ ?

• Go back to the basic definition of an eigenvector and try it. – amd Nov 3 '16 at 2:28
• Try the identity matrix... – user251257 Nov 3 '16 at 2:43
• @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 ? – Widawensen Nov 3 '16 at 9:57
• @Widawensen I mean that if they are both eigenvalues is the addition also an eigenvalue – SFL Nov 3 '16 at 11:07
• @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) – Widawensen Nov 3 '16 at 12:24

Not necessarily: if $T v_1 = \lambda_1 v_1$ and $T v_2 = \lambda_2 v_2$ then
$$T(v_1 + v_2) = \lambda_1 v_1 + \lambda_2 v_2 \ne (\lambda_1 + \lambda_2)(v_1 + v_2)$$