Productive matrix

In linear algebra, a square nonnegative matrix

is said to be productive, or to be a Leontief matrix, if there exists a

nonnegative column matrix

is a positive matrix.

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.

[1] The interdependency linkages between the latter can be examined by the input-output model with empirical data.

denotes the set of r×c matrices of real numbers, whereas

indicates a positive and a nonnegative matrix, respectively.

The following properties are proven e.g. in the textbook (Michel 1984).

[2] Theorem A nonnegative matrix

is invertible with a nonnegative inverse, where

identity matrix.

Proof "If" : "Only if" : Proposition The transpose of a productive matrix is productive.

Proof 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.