Matrix Representation of Planar Lorentz Transformations in Extended (2 + 2)-Dimensional Spacetime

A matrix formulation of Lorentz transformations restricted to planar motion is developed within an extended 2+2-dimensional spacetime framework. Focusing on inertial frames in relative motion confined to the XY-plane, the analysis begins with radial Lorentz boosts, where the relative velocity is collinear with the spatial position vector. By decomposing the temporal coordinate into components aligned with the plane, radial two-dimensional Lorentz transformations are naturally embedded into a four-component formalism with metric signature +,+,−,−. This construction enables a compact matrix representation of planar radial boosts that preserves the spacetime interval and admits a well-defined inverse obtained by velocity reversal. The approach is then generalized to arbitrary boost directions within the plane. Explicit transformation matrices for arbitrary planar Lorentz boosts are then derived in block form, highlighting the symmetric coupling between spatial and temporal-like components. The resulting 2+2-dimensional matrices are shown to preserve the extended metric, thereby establishing Lorentz invariance for planar motion and providing a unified, compact, and algebraically transparent description of planar relativistic kinematics.