Dimension of Eigenspace of A and A^T

Thread starter potassium_mn04
Start date May 2, 2021
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Dimension

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The discussion centers on the relationship between the dimensions of eigenspaces of a matrix A and its transpose A^T, emphasizing that the ranks of both matrices are equal. The eigenspace corresponding to an eigenvalue λ is defined as the nullspace of λI - A, with its dimension being the nullity of this expression. Participants suggest applying the rank-nullity theorem to establish the connection between the dimensions of the eigenspaces. Understanding this theorem is crucial for proving the equality of the dimensions. Overall, the conversation highlights the importance of the rank-nullity theorem in linear algebra.

May 2, 2021

potassium_mn04

Homework Statement: Prove that the dimension of the eigenspace of A and dimension of eigenspace of A^T are equal

Relevant Equations: dim(E_A)=dim(E_(At))

I know that the rank of A and A^T are equal, and that the statement follows from there, but I have no idea how to prove it.

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May 2, 2021

S.G. Janssens

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The eigenspace of ##A## corresponding to an eigenvalue ##\lambda## is the nullspace of ##\lambda I - A##. So, the dimension of that eigenspace is the nullity of ##\lambda I - A##. Are you familiar with the rank-nullity theorem? (If not, then look it up: Your book may call it differently.) You can apply that theorem here.

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