Question

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

Answer 1

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 `mathbf e` is an eigenvector of both of these matrices, then we can say that:

• `M_1 mathbf e = lambda_1 mathbf e`; and

• `M_2 mathbf e = lambda_2 mathbf e`

....where `lambda_1` and `lambda_2` are the associated eigenvalues .

It follows from adding these together that:

`M_1 mathbf e + M_2 mathbf e = lambda_1 mathbf e+ lambda_2 mathbf e`

`implies (M_1 + M_2) mathbf e = (lambda_1 + lambda_2) mathbf e`

`implies M_3 mathbf e = lambda_3 mathbf e`

Where: `M_3 = M_1 + M_2`