When is there an inner product making a given matrix unitary?

Question

The unitarity of a given operator/matrix $A$ depends on the underlying inner product, so I guess it in principle possible for a given non-unitary $A$ to act as a unitary with respect to a different choice of inner product.

Is this ever possible? If yes, are there necessary and sufficient conditions on $A$ that make it unitary with respect to some inner product?

Answer 1

A diagonalizable matrix with all eigenvalues of modulus 1 is "unitarizable" by selecting an inner product in which its eigenvectors with distinct eigenvalues are all mutually orthogonal.