Triangular Decomposition of the Admittance Matrix, Gauss Elimination Method and the General Star-Mesh Transformation

The star-mesh transformation allows the reduction of nodes in an electrical circuit.
The Gauss elimination method allows the reduction of unknowns in a system of linear
Equations . Triangular decomposition can be used to find the inverse of a matrix.
In this paper the coherence between these methods are discussed. It is shown
that the Gauss elimination method gives the same formula for star-mesh transformation.
It is further shown that the element B of the upper triangular matrix B obtained by triangular decomposition of the admittance matrix G is the admittance between the nodes 2 and K if the node I is eliminated. And the element B is the admittance
1J between the nodes i and j if the nodes 1 through i-1are eliminated . Similarly the element B of the upper triangular matrix B obtained by decomposition of the inductance matrix is the mutual inductance of the coils i and j if the coils 1 through i-1 are short circuited. The element C of the corresponding lower triangular matrix, is the noload voltage ratio of the coils i and k if the coils 1 through i-1 are short circuited.