  We prove an upper bound on the total variation mixing time of a finite Markov
chain in terms of the absolute spectral gap and the number of elements in the
state space. Unlike results requiring reversibility or irreducibility, this
bound is finite whenever the chain converges. The dependence on the number of
elements means that the new bound cannot capture the behavior of rapidly mixing
chains; but an example shows that such dependence is necessary. We also provide
a sharpened result for reversible chains that are not necessarily irreducible.
The proof of the general bound exploits a connection between linear recurrence
relations and Schur functions due to Hou and Mu, while the sharpened bound for
reversible chains arises as a consequence of the stochastic interpretation of
eigenvalues developed by Diaconis, Fill, and Miclo. In particular, for every
reversible chain with nonnegative eigenvalues $\beta_j$ we find a strong
stationary time whose law is the sum of independent geometric random variables
having mean $1/(1-\beta_j)$.
