Could it be that the action of a Markov Chain on vectors, where the vector converges to an eigenvector, is similar to the action of an LLM under gradient descen

Could it be that the action of a Markov Chain on vectors, where the vector converges to an eigenvector, is similar to the action of an LLM under gradient descent, where vectors converge to a minimum? There's also a link between eigenvectors and solutions to simultaneous differential equations... I'm not that knowledgeable about all of this but it feels like there are many similarities.