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

1 Answer
Eddie
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_1M1 and M_2M2, and mathbf e e is an eigenvector of both of these matrices, then we can say that:

  • M_1 mathbf e = lambda_1 mathbf eM1e=λ1e; and

  • M_2 mathbf e = lambda_2 mathbf eM2e=λ2e

....where lambda_1λ1 and lambda_2λ2 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 eM1e+M2e=λ1e+λ2e

implies (M_1 + M_2) mathbf e = (lambda_1 + lambda_2) mathbf e⇒(M1+M2)e=(λ1+λ2)e

implies M_3 mathbf e = lambda_3 mathbf e⇒M3e=λ3e

Where: M_3 = M_1 + M_2M3=M1+M2

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