  For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several
classes of matrices from $A$ for which the conjecture will also hold. We
discover a new link between cyclicity and Crouzeix's conjecture, which shows
that Crouzeix's Conjecture holds in full generality if and only if it holds for
the differentiation operator on a class of analytic functions. We pose several
open questions, which if proved, will prove Crouzeix's conjecture. We also
begin an investigation into Crouzeix's conjecture for symmetric matrices and in
the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for
symmetric matrices if and only if it holds for analytic truncated Toeplitz
operators.
