Reduced incidence matrix

Length of ith pipe section (95) Reduced incidence matrix. M refers lo a digraph (7), Wl refers to an undirected graph (12), and M refers to a digraph containing only internal edges (34)... 

Note that junctions fall into the category of non-mixing, reversible transducers. They may be seen as a TF or MTF with a constitutive matrix that contains only 1,-1, and 0 as matrix elements and modulation consists of changing the absolute values of this matrix. Furthermore, the same holds for multiport substructures that only contain junctions. The transpose of this matrix relates the independent voltages to the dependent ones and thus corresponds to those columns of the reduced incidence matrix of an electrical circuit that relate the link voltages to the branch voltages. 

This makes clear that any two electrical terminals, also if one is grounded and not represented as such, form a port. However, in many treatments of electrical circuits the logical order is reversed First, a distinction is made between potential and potential differences. Next a relation between the potential differences and the potentials is derived via the incidence matrix and only then it is recognized that one of the rows (balance equations) refers to a reference node (ground) which should be omitted from the incidence matrix to obtain the so-called reduced incidence matrix. This culture may have led Willems to drawing this conclusion. 

When restricting the graph G to arcs j g J , we obtain a subgraph (say) G° its reduced incidence matrix is of elements C j where n N and j g J . The subgraph is generally not connected it can even contain isolated nodes, not incident with any arc j g J°. Example ... 

Let us arrange the subgraphs G° in the manner that G , , G - K < K) are those subgraphs which are not isolated nodes observe that if G is an isolated node, the corresponding (scalar) equation (node balance) in (3.2.2) becomes automatically one of the node balances of the reduced graph G. Having selected a reference node in each G for k = I, , K , let B be the reduced incidence matrix of G, thus B is of full row rank let further A, be the corresponding... 

Having completed the classification according to steps (a) and (b) above, we can make use of the additional information obtained in the described manner. First, the reduced graph G determines, having selected a reference node, the reduced incidence matrix A. It is the matrix occurring in (3.2.4)2, thus in the constraint equation for the measured vector m (in fact, only for the subvector m of redundant variables). The equation is employed for adjusting the given values if the components of m"" have been actually measured then for reconciliation by statistical methods. 

The inverse R2 of the reduced incidence matrix is found as in the example at the end of Section A.3 for each node n e N2, the reference node excluded, we go backwards through the sequence of predecessors. We have... 

In addition, the incidence matrix of G determines the conditions the measured variables have to obey in order to have the system solvable. Taking node , as reference node, the reduced incidence matrix equals (cf. Fig. 3-9c)... 

If only mass is balanced we can formally admit a change in accumulation of mass in a node as a fictitious stream oriented towards die environment see Fig. 3-1. The conservation of mass is then expressed as Eq.(3.1.6) thus Cm = 0 where m is the (column) vector of mass flowrates and C the reduced incidence matrix of G, thus without row Hq. We assume G connected, hence C is of full row rank (3.1.5) thus N where M denotes generally the number of elements of set M. 

The matrix C of elements C j (n e N , 7 e J) is the reduced incidence matrix of graph G, with the environment node as reference node see Chapter 3. Let us now admit the presence of energy distributors. Then the matrix D of... 

Then, in (5.2.4), C is reduced incidence matrix of the modified mass flow graph restricted to arcs j e J . Matrix C consists of elements C j where n e T and... 

As a simplifying but plausible hypothesis, we can assume that the subgraph G [N-S, ] of reduced incidence matrix C is connected, thus C is of full row... 

Consider in particular the system of mass balance equations (Chapter 3). Then the reduced incidence matrix C of graph G in (3.1.6) is partitioned... 

Possibly unmeasured are only certain components of the subvectors 1%, say Wji for / e J = JfJf. TTien the subgraph G° of G restricted to unmeasured streams is of reduced incidence matrix (say) B. We thus partition... 

Observe that Cf is reduced incidence matrix of the graph (say) Gf [N, Jf ] representing the system of given nodes and of material streams, disregarding the accumulation. It is quite natural to assume that also... 

Let us begin with the case that G is a tree, let A, = A be the reduced incidence matrix. It is well-known that A is then square by (A.3), and regular. We shall give an expression to the inverse. 

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