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

edit

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.

Explicit definition

edit

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

edit

The following properties are proven e.g. in the textbook (Michel 1984).[2]

Characterization

edit

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

Proof

"If" :

Let   be invertible with a nonnegative inverse,
Let   be an arbitrary column matrix with  .
Then the matrix   is nonnegative since it is the product of two nonnegative matrices.
Moreover,  .
Therefore   is productive.

"Only if" :

Let   be productive, let   such that  .
The proof proceeds by reductio ad absurdum.
First, assume for contradiction   is singular.
The endomorphism canonically associated with   can not be injective by singularity of the matrix.
Thus some non-zero column matrix   exists such that  .
The matrix   has the same properties as  , therefore we can choose   as an element of the kernel with at least one positive entry.
Hence   is nonnegative and reached with at least one value  .
By definition of   and of  , we can infer that:
 
 , using that   by construction.
Thus  , using that   by definition of  .
This contradicts   and  , hence   is necessarily invertible.
Second, assume for contradiction   is invertible but with at least one negative entry in its inverse.
Hence   such that there is at least one negative entry in  .
Then   is positive and reached with at least one value  .
By definition of   and of  , we can infer that:
 
 , using that   by construction
  using that   by definition of  .
Thus  , contradicting  .
Therefore   is necessarily nonnegative.

Transposition

edit

Proposition The transpose of a productive matrix is productive.

Proof

Let   a productive matrix.
Then   exists and is nonnegative.
Yet  
Hence   is invertible with a nonnegative inverse.
Therefore   is productive.

Application

edit

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.

References

edit
  1. ^ Kim Minju, Leontief Input-Output Model (Application of Linear Algebra to Economics) Archived 2014-12-15 at the Wayback Machine
  2. ^ Philippe Michel, "9.2 Matrices productives", Cours de Mathématiques pour Economistes, Édition Economica, 1984