  We study thermalization in closed non-integrable quantum systems using the
Krylov basis. We demonstrate that for thermalization to occur, the matrix
representation of typical local operators in the Krylov basis should exhibit a
specific tridiagonal form with all other elements in the matrix are
exponentially small, reminiscent of the eigenstate thermalization hypothesis.
Within this framework, we propose that the nature of thermalization, whether
weak or strong, can be examined by the infinite time average of the Krylov
complexity. Moreover, we analyze the variance of Lanczos coefficients as
another probe for the nature of thermalization. One observes that although the
variance of Lanczos coefficients may capture certain features of
thermalization, it is not as effective as the infinite time average of
complexity.
