Eigenvarieties are parameter spaces for certain p-adic automorphic forms of varying weight. These objects have become increasingly popular for studying the Fontaine—Mazur conjecture, which leads us to ask what kinds of Galois representations appear on eigenvarieties. Our main result shows that for eigenvarieties for the group GL_n over a CM field, the associated Galois representations are trianguline at all p-adic places, resolving a conjecture of Hansen (following Kisin, Colmez, Bellaiche—Chenevier). The strategy of proof (which could be of independent interest) is to embed eigenvarieties for GL_n into an eigenvariety for a 2n-variable unitary group.