Productive matrix

In linear algebra, a square nonnegative matrix A {displaystyle A} of order n {displaystyle n} is said to be productive, or to be a Leontief matrix, if there exists a n × 1 {displaystyle n imes 1} nonnegative column matrix P {displaystyle P} such as P − A P {displaystyle P-AP} is a positive matrix. In linear algebra, a square nonnegative matrix A {displaystyle A} of order n {displaystyle n} is said to be productive, or to be a Leontief matrix, if there exists a n × 1 {displaystyle n imes 1} nonnegative column matrix P {displaystyle P} such as P − A P {displaystyle P-AP} 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. The interdependency linkages between the latter can be examined by the input-output model with empirical data. The matrix A ∈ M n , n ( R ) {displaystyle Ain mathrm {M} _{n,n}(mathbb {R} )} is productive if and only if A ⩾ 0 {displaystyle Ageqslant 0} and ∃ P ∈ M n , 1 ( R ) , P > 0 {displaystyle exists Pin mathrm {M} _{n,1}(mathbb {R} ),P>0} such as P − A P > 0 {displaystyle P-AP>0} . Here M r , c ( R ) {displaystyle mathrm {M} _{r,c}(mathbb {R} )} denotes the set of r×c matrices of real numbers, whereas > 0 {displaystyle >0} and ⩾ 0 {displaystyle geqslant 0} indicates a positive and a nonnegative matrix, respectively. The following properties are proven e.g. in the textbook (Michel 1984). TheoremA nonnegative matrix A ∈ M n , n ( R ) {displaystyle Ain mathrm {M} _{n,n}(mathbb {R} )} is productive if and only if I n − A {displaystyle I_{n}-A} is invertible with a nonnegative inverse, where I n {displaystyle I_{n}} denotes the n × n {displaystyle n imes n} identity matrix. Proof