Fundamental properties of the main matrix forms in the systems of equations of intersectoral balance

Kulyk М.M., Academician of the National Academy of Sciences of Ukraine, Dr. Sci. (Eng.), Professor, https://orcid.org/0000-0002-5582-7027
Institute of General Energy of the National Academy of Sciences of Ukraine, 172 Antonovycha str., Kyiv, 03150, Ukraine
Language: Russian, English
Source: The Problems of General Energy, 2017, 2(49):14-39
https://doi.org/10.15407/pge2017.02.014
Section: Mathematical modeling of energy facilities and systems
UDC: 622.324
Received: 05.06.2017
Published: 14.07.2017

Abstract:

We have investigated the fundamental properties of the matrices of three systems of algebraic equations describing the key problems of intersectoral balance: determination of output by the data of final demand, determination of output by the data of added value, and establishing the interrelation between equilibrium prices and output. All these problems have been solved with using the same method proposed here and called the method of extrapolation to zero determinant. We have shown that the system of equations recommended by numerous authors for establishing the interrelation between equilibrium prices and output volumes in output units is homogeneous. We have proved that, in this matrix, there exist at least n positive minors with a dimension r = n - 1 , where n is the dimension of matrix. This important feature envisions the fact that, in the continual set of solutions of the corresponding system, it is impossible to find even if a single vector that would correspond to the meaning content of tables “input-output”. Therefore, this system cannot be used not only for determining equilibrium prices and output in output units, but also for the solution of other problems of intersectoral balance.
We have proved that the matrix of the system of equations for finding output by the data of final demand has a rank r = n, and its determinant is always positive for any dimension of this matrix and non-negativeness of the elements of final demand, and the last condition not always is necessary.
We have established that the system of equations for finding output by the data of added value has a matrix whose rank is  r = n, and its determinant is positive for all values of variables satisfying the condition of input balance. The property of positiveness of the determinant of this matrix was already proved by R. Bellman, and this proof was based on the Gram determinants. It is more compact than the method of extrapolation to zero determinant, but the latter can be applied to all problems considered here and, which is quite important, enables one to obtain not only the sign, but also the values of determinants under study.
We should especially emphasize the unique property of matrices of the systems of equations for determining output by the data of final demand and added value. The determinants of these matrices depend on the values of right-hand sides of the corresponding systems, namely, on the values of elements of the vectors of final demand and added value, respectively. The initial cause of such property is the fact that these systems were constructed on the basis of output and input balances.

Keywords: matrix, determinant, rank, vector, output, input, balance, price.

References:

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