This chapter studies unitary matrices and contractive matrices (contractions). Sect. 6.1 gives basic properties of unitary matrices, Sect. 6.2 studies the structures of real orthogonal matrices under similarity, Sect. 6.3 discusses the connections of contractions with unitary matrices, Sect. 6.4 concerns the unitary similarity of real matrices, and Sect. 6.5 presents a trace inequality for unitary matrices.