A criterion for a contraction T on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function \Theta_{T} of T in 2007 by Chevrot, Fricain, and D. Timotin. To further classify unitary equivalent complex symmetric contractions, we notice a simple condition of when \Theta_{T_{1}} and \Theta_{T_{2}} coincide for two complex symmetric contractions T_{1} and T_{2}. As an application, surprisingly we solve the problem for any defect index n, when the defect indexes of contractions are 2, this problem was left open in them. Furthermore, a construction of 3\times3 symmetric inner matrices is proposed, which extends some results on 2\times2 inner matrices by Garcia and 2\times2 symmetric inner matrices by Chevrot, Fricain, and D. Timotin.

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