  We study geodesics of multivariate normal distributions with respect to the
Fisher metric. First it will be shown that a computational formula for
geodesics can be understood using the block Cholesky decomposition and a
natural Riemannian submersion. Next a mid point algorithm for geodesics will be
obtained. And finally a new Toda lattice type Lax pair will be derived from the
geodesic and the block Cholesky decomposition.
