Are these two things related somehow in an intuitive way?
2026-04-29 21:45:08.1777499108
Relationship between $A x=b$ and $A x=\lambda x$
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It depends; do you know $\lambda$? If so, you can reduce the second problem to a linear system by rearranging to get
$$(A-\lambda I)x = 0.$$
which is of the same form as your first problem.
If $\lambda$ is unknown and you are trying to simultaneously find $\lambda$ and $x$, then the second equation is called an eigenvector problem and is not equivalent to solving any single linear system.
Numerical techniques exist for finding eigenvectors, though they are both more complex and more expensive than linear solvers; some keywords to get you started include "spectral decomposition," "singular value decomposition," and "power iteration."