Krawtchouk matrices from the Feynman path integral and from the split quaternions

An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.