Linear Algebra in the vector space of intervals

In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with respect to the real case, a matrix of order n could have more than n eigenvalues (the set of intervals is not factorial). We consider a notion of central eigenvalues permits to describe criterium of diagonalization. As application, we define a notion of Exponential mapping. 1 The associative algebra IR In [1], we have given a representation of the set of intervals in terms of associative algebra. More precisely, we define on the set IR of intervals of R a R-vector space structure. Next we embed IR in a 4-dimensional associative algebra. This embedding permits to describe a unique distributive multiplication which contains all the possible results of the usual product of intervals and the monotony property is always conserved. Moreover, this new product is minimal with respect the distributivity and the monotony properties. In this section, we present briefly this construction (for more details, see [1]). 1.1 Vector space structure on IR Let IR be the set of intervals of R, that is IR = {[a, b], a, b ∈ R, a ≤ b}. This set is provided with a semi-group structure that we can complete as follow: we consider the equivalence relation on IR× IR: (X,Y ) ∼ (Z, T ) ⇐⇒ X + T = Y + Z for all X,Y, Z, T ∈ IR. The quotient set is denoted by IR. The addition of intervals is compatible with this equivalence relation : (X,Y ) + (z, t) = (x + z, y + t) where (x, y) is the equivalence class of (X,Y ). The unit is 0 = {(X,X), X ∈ IR} and each element has an inverse r(X,Y ) = (Y,X).