I am new to linear algebra but I have been trying to figure out this question. Everybody seems to take for granted that for matrix A which has eigenvectors x, A(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}also has the same eigenvectors?

I know that people are just operating on the equation Ax=λx, saying that A^{2}x=A(Ax)=A(λx) and therefore A^{2}x = λ^{2}x. However, in my opinion, this is not a proof proving why A^{2}and A have the same eigenvectors but rather why λ is squared on the basis that the matrices share the same eigenvectors.

If someone can prove that A^{2}and A have the same eigenvectors by using equations A^{2}y=αy and Ax=λx, and proceeding to prove y=x, I will be very much convinced that these two matrices have the same eigenvectors.

Or are there any other convincing proofs to show this result?

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# I Why do eigenvectors stay the same when a matrix is squared?

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