# Gram matrix in a non-Euclidean space

The Gram matrix of a set of vectors $v_1,\dots,v_n$ with the usual Euclidean dot product is positive semidenifite.

Suppose we have a symmetric matrix $A$ that is not positive semidefinite. Can we interpret this matrix as the Gram matrix in some non-Euclidean space (such as the hyperbolic space)? If so, on what conditions?

• I like Lattices and Codes by Wolfgang Ebeling. That's where I learned the material. I do not know any text that uses Lorentz metrics on lattices from the very start. If you are ambitious, SPLAG by Conway and Sloane, with chapters by Borcherds. Sphere Packings, Lattices and Groups. – Will Jagy Jun 11 '18 at 2:53
• Thank you for the pointers. I'll check the books you mentioned. – pharmine Jun 11 '18 at 11:22