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

Geodesics of multivariate normal distributions and a Toda lattice type Lax pair