Productive matrix

In linear algebra, a square nonnegative matrix of order is said to be productive, or to be a Leontief matrix, if there exists a nonnegative column matrix such as is a positive matrix.

History

The concept of productive matrix was developed by the economist Wassily Leontief 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 is productive if and only if and such as.
Here denotes the set of r×''c'' matrices of real numbers, whereas and indicates a positive and a nonnegative matrix, respectively.

Properties

The following properties are proven e.g. in the textbook.

Characterization

Theorem
A nonnegative matrix is productive if and only if is invertible with a nonnegative inverse, where denotes the 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.