We present a new numerical abstract domain based on ellip- soids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the diffi- culty that ellipsoids do not have a lattice structure by ex- hibiting a canonical operator over-approximating the union. This operator is the only one which permits to perform anal- yses that are invariant with respect to a linear transforma- tion of state variables. Moreover, we show that this operator can be computed efficiently using basic algebraic operations on positive semidefinite matrices. We finally develop a fast non-linear power-type algorithm, which allows one to de- termine sound quadratic invariants on switched systems in a tractable way, by solving fixed point problems over the space of ellipsoids. We test our approach on several bench- marks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances.