  A real Bott manifold is the total space of a sequence of $\R P^1$ bundles
starting with a point, where each $\R P^1$ bundle is projectivization of a
Whitney sum of two real line bundles. A real Bott manifold is a real toric
manifold which admits a flat riemannian metric. An upper triangular $(0,1)$
matrix with zero diagonal entries uniquely determines such a sequence of $\R
P^1$ bundles but different matrices may produce diffeomorphic real Bott
manifolds. In this paper we determine when two such matrices produce
diffeomorphic real Bott manifolds. The argument also proves that any graded
ring isomorphism between the cohomology rings of real Bott manifolds with
$\Z/2$ coefficients is induced by an affine diffeomorphism between the real
Bott manifolds. In particular, this implies the main theorem of \cite{ka-ma08}
which asserts that two real Bott manifolds are diffeomorphic if and only of
their cohomology rings with $\Z/2$ coefficients are isomorphic as graded rings.
We also prove that the decomposition of a real Bott manifold into a product of
indecomposable real Bott manifolds is unique up to permutations of the
indecomposable factors.
