The causative-matrix method to analyze temporal change assumes that a matrix transforms one Markovian transition matrix into another by a left multiplication of the first matrix; the method is demand-driven when applied to input-output economics. An extension is presented without assuming the...

Biproportional methods are used to update matrices: the projection of a matrix Z to give it the column and row sums of another matrix is R Z S, where R and S are diagonal and secure the constraints of the problem (R and S have no signification at all because they are not identified). However,...

Biproportional methods project a matrix <Emphasis Type="Bold">A to give it the column and row sums of another matrix; the result is <Emphasis Type="Bold">R A S, where <Emphasis Type="Bold">R and <Emphasis Type="Bold">S are diagonal matrices. As <Emphasis Type="Bold">R and <Emphasis Type="Bold">S are not identified, one must normalize them, even after computing, that is, ex post. This article starts from the idea developed in...</emphasis></emphasis></emphasis></emphasis></emphasis></emphasis>