given matrix A, for eigen vector of A, V1 (coordinates in base B), with eigen value X in the field - is V1 by base B' is still an eigen vector of A with the SAME eigen value X?
If so, can you show me how come?
2026-03-30 10:54:37.1774868077
Eigen Vector with eigen value in different bases
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Indicating with $M$ the matrix for the change of basis we have that $V_1=MW_1$ and then
$$AV_1=XV_1\implies AMW_1=XMW_1\implies M^{-1}AMW_1=XW_1\implies BW_1=XW_1$$
thus the eigenvalue is $X$ but the representation of $V_1$ changes according to the new basis.