H-Unitary and Lorentz Matrices: A Review

Abstract

Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices that are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix are called Lorentz. The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H-unitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and H-unitary, products of reflections, and stability and robust stability.