Productive matrix

In linear algebra, a square nonnegative matrix formula_1 of order formula_2 is said to be productive, or to be a Leontief matrix, if there exists a formula_3 nonnegative column matrix formula_4 such as formula_5 is a positive matrix.
History.
The concept of productive matrix was developed by the economist Wassily Leontief (Nobel Prize in Economics in 1973) in order to model and analyze the relations between the different sectors of an economy. The interdependency linkages between the latter can be examined by the input-output model with empirical data.
Explicit definition.
The matrix formula_6 is productive if and only if formula_7 and formula_8 such as formula_9.
Here formula_10 denotes the set of "r"×"c" matrices of real numbers, whereas formula_11 and formula_12 indicates a positive and a nonnegative matrix, respectively.
Properties.
The following properties are proven e.g. in the textbook (Michel 1984).
Characterization.
Theorem
A nonnegative matrix formula_6 is productive if and only if formula_14 is invertible with a nonnegative inverse, where formula_15 denotes the formula_16 identity matrix.
Proof
"If" :
"Only if" :
Transposition.
Proposition
The transpose of a productive matrix is productive.
Proof
Application.
With a matrix approach of the input-output model, the consumption matrix is productive if it is economically viable and if the latter and the demand vector are nonnegative.