  The matrix-vector wave equation is a compact first-order differential
equation. It was originally used for the analysis of elastodynamic plane waves
in laterally invariant media. It has been extended by various authors for
laterally varying media. Other authors derived a similar formalism for other
wave phenomena. This paper starts with a unified formulation of the
matrix-vector wave equation for 3D inhomogeneous, dissipative media. The wave
vector, source vector and operator matrix are specified in the appendices for
acoustic, quantum mechanical, electromagnetic, elastodynamic,
poroelastodynamic, piezoelectric and seismoelectric waves. It is shown that the
operator matrix obeys unified symmetry relations for all these wave phenomena.
Next, unified matrix-vector reciprocity theorems of the convolution and
correlation type are derived, utilising the symmetry properties of the operator
matrix. These theorems formulate mathematical relations between two wave states
in the same spatial domain. A unified wave field representation is obtained by
replacing one of the states in the convolution-type reciprocity theorem by a
Green's state. By replacing both states in the correlation-type reciprocity
theorem by Green's states, a unified representation of the homogeneous Green's
matrix is obtained. Applications of the unified reciprocity theorems and
representations for forward and inverse wave problems are briefly indicated.
