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A matrix G has eigen vector e How can i prove that e is also an eigen vector for G+k I where I is an identity matrix. (they are all 3x3). I tried using Ae = k e i didnt work

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Just applying the definition one has

$$(\mathbf{G}+k\mathbf{I})v=\lambda v+k v=(\lambda+k)v$$

Which means $v$ is an eigenvector of $\mathbf{G}+k\mathbf{I}$ associated to the eigenvalue $\lambda+k$

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