Why Do Eigenvectors Stay The Same When A Matrix Is Squared?

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UndergradWhy do eigenvectors stay the same when a matrix is squared?

• Thread starter Thread starter Aldnoahz
• Start date Start date Sep 11, 2016
• Tags Tags Eigenvectors Matrix

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The discussion clarifies that while a matrix A and its square A² share some eigenvectors, they do not necessarily have the same set of eigenvectors. The proof provided demonstrates that if Ax = λx, then A²x = λ²x, confirming that eigenvectors of A remain eigenvectors of A². However, it is established that A² can have additional eigenvectors not present in A, as illustrated by the example of the derivative operator D, where D² has eigenvector x, but D does not. Thus, the relationship between eigenvectors of A and A² is not one-to-one.

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• Understanding of linear algebra concepts, specifically eigenvalues and eigenvectors.
• Familiarity with matrix operations, including matrix multiplication.
• Knowledge of the definitions and properties of linear transformations.
• Basic understanding of polynomial functions and their derivatives.

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• Study the properties of eigenvalues and eigenvectors in depth.
• Learn about linear transformations and their effects on vector spaces.
• Explore examples of matrices with distinct eigenvectors and their squares.
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This discussion is beneficial for students of linear algebra, mathematicians exploring eigenvalue problems, and educators seeking to clarify the relationship between a matrix and its square in terms of eigenvectors.

Aldnoahz Messages 37 Reaction score 1 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, A2 also has the same eigenvectors? I know that people are just operating on the equation Ax=λx, saying that A2x=A(Ax)=A(λx) and therefore A2x = λ2x. However, in my opinion, this is not a proof proving why A2 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 A2 and A have the same eigenvectors by using equations A2y=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? Physics news on Phys.org

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fresh_42 Staff Emeritus Science Advisor Homework Helper Insights Author 2024 Award Messages 20,815 Reaction score 28,439 You already have said everything that can be said here. I don't know why you don't regard it as a proof. There is no way of proving ##A^2y=\alpha y \; \wedge \; Ax= \lambda x \; \Rightarrow \; x=y## because it is not true. E.g. ##x## and ##y## can simply be two different (linear independent) eigenvectors of ##A##. What you can prove is ##Ay=\alpha y \; \Rightarrow \; A^2y=\alpha^2 y## which is done by the equation you posted. FactChecker Science Advisor Homework Helper Messages 9,360 Reaction score 4,638

Aldnoahz said: therefore A2x = λ2x.

Let B = A2 and α = λ2. Then Bx = αx. That is the definition of x being an eigenvector of B. Aldnoahz Messages 37 Reaction score 1 Yeah I think I had some problems with my logic. Now I understand. Thank you. mathwonk Science Advisor Homework Helper 2024 Award Messages 11,959 Reaction score 2,232 i am not sure what you have concluded but it is not true that A^2 has the same eigenvectors as A, since it can have more. E.g. take D the derivative acting on polynomials of degree ≤ one. Then D^2 = 0 and thus has x as an eigenvector, since D^2x = 0, but D does not since Dx = 1. Of course an eigenvector of A is also an eigenvector of A^2, "trivially", as proved above, but the converse is false. Last edited: Sep 17, 2016

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