Maxwell-Lorentz Matrix

Lorentz invariance of Maxwell electromagnetic equations is demonstrated in two complementary ways: first, we give a pedestrian review with three-vector equations, and we then express Maxwell equations in a four-vector matrix form (the Maxwell-Lorentz matrix) which demonstrates the intimate connection of Maxwell equations with the Lorentz group. Each Maxwell-Lorentz matrix component is the product of three matrices: a derivative matrix, a 4x4 Lorentz group generator matrix, and an electromagnetic field matrix. We obtain rotary Lorentz transformations of the electromagnetic field matrix from Lorentz equation matrices. We then transform the derivative and electromagnetic matrices and obtain an explicit matrix demonstration of Lorentz invariance of Maxwell equations. To obtain this result, we express all transformation matrices in exponential form to facilitate the application of simple Lorentz group algebra. The pedestrian approach illustrates what the Lorentz group matrix approach actually accomplishes and helps one to gain some appreciation of group theory methods.