# Is the set of all 3 × 3 matrices that have the vector [2, 1 , -2]^T as an eigenvector closed under addition?

Mar 31, 2017

Yes, see below.

#### Explanation:

A set is closed under addition if the sum of any two elements in the set is also in the set.

Here, if 2 matrices in the set are ${M}_{1}$ and ${M}_{2}$, and $m a t h b f e$ is an eigenvector of both of these matrices, then we can say that:

• ${M}_{1} m a t h b f e = {\lambda}_{1} m a t h b f e$; and

• ${M}_{2} m a t h b f e = {\lambda}_{2} m a t h b f e$

....where ${\lambda}_{1}$ and ${\lambda}_{2}$ are the associated eigenvalues .

It follows from adding these together that:

${M}_{1} m a t h b f e + {M}_{2} m a t h b f e = {\lambda}_{1} m a t h b f e + {\lambda}_{2} m a t h b f e$

$\implies \left({M}_{1} + {M}_{2}\right) m a t h b f e = \left({\lambda}_{1} + {\lambda}_{2}\right) m a t h b f e$

$\implies {M}_{3} m a t h b f e = {\lambda}_{3} m a t h b f e$

Where: ${M}_{3} = {M}_{1} + {M}_{2}$