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In this paper, we study matrix representations of truncated Toeplitzoperators with respect to orthonormal bases which are invariant under acanonical conjugation map. In particular, we determine necessary and sufficientconditions for when a 3-by-3 symmetric matrix is the matrix representation of atruncated Toeplitz operator with respect to a given conjugation invariantorthonormal basis. We specialise our result to the case when the conjugationinvariant orthonormal basis is a modified Clark basis. As a corollary to thisspecialisation, we answer a previously stated open conjecture in the negative,and show that not every unitary equivalence between a complex symmetric matrixand a truncated Toeplitz operator arises from a modified Clark basisrepresentation. Finally, we show that a given 3-by-3 symmetric matrix is thematrix representation of a truncated Toeplitz operator with respect to aconjugation invariant orthonormal basis if and only if a specified system ofpolynomial equations is satisfied with a real solution.

Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces