"Linear algebra" graph and Network

The previous articles on linear algebra are all explained in terms of mathematics. This article will be a different angle to explain the problem.

The tutor often tells me that everything has to think about its physical or practical significance, through the phenomenon to see the essence, so that can be more profound understanding, so you can see what the actual use of linear algebra.

If there are for example the following circuit networks:

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There are 1,2,3,4 nodes in the figure, the y1,y2,y3,y4,y5 five edges, and the arrows pointing to indicate the current flow.

If the starting point of the current is set to-1, the arrival point is set to 1. we are able to represent these networks through matrices:

The physical meaning of matrix 0 space

We first consider the 0 space of matrix a . Then there are:

From the above, it can be seen that the result of Ax is the difference between the nodes , if Xi is the first point of the potential, then we give the physical meaning: ax=0 means that when the node potential value, the potential difference between all nodes is 0. Very clearly. When all the nodal potentials are clearly established, namely:

We will show the current of each side of the graph in B, then there is the equation ax=b. A indicates the relationship between the circuit nodes, X represents the potential of each point, and B represents the current on each side.

this way. We mathematically set a physical problem .

The above discussed in the b=0,x of the components of the same, that is, the electric potential of the same, there is no current in the network, which is consistent with our physical knowledge: Potential difference is the cause of the current generation.

The physical meaning of the matrix left 0 space
above we look at the 0 space of matrix A, below we discuss the left 0 space of matrix a , in order to give practical meaning to our equation. In the following formula. If Y is the current for each edge, B is the current value for each node, and the b=0 indicates that the current is 0.

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the type of reaction is the Kirchhoff current law in the current : the current flowing into a node is equal to the outflow, i.e. the current is 0

For example,-y1-y3-y4=0 indicates that the 1th node satisfies the KCL condition.

we can obtain the solution by the Gaussian elimination method, but we can get the solution by the graph: in the circuit Network diagram, we can see that the number of the node is a loop. The 1,3,4 also forms a loop in order to satisfy the Kirchhoff current law. We can just let the current flow through the loop. That is: we can get two special solutions:

Then the left 0 space can be expressed as a linear combination of the above two special solutions.

Eliminate the element for at. We can find that its rank is 3, that is, three rows in the column are linearly independent. by the above two special solutions, we know the 1th one. 2. The 3 columns are relevant, 3rd. 4, 5 columns are relevant, except that the arbitrary three columns are linearly independent . We found that the Y1,y2,y3 exactly formed the loop. The y3,y4,y5 also happens to form loops, indicating that the correlation is generated by the loop.
Proof of Euler's formula
Euler formula: number of loops = number of sides-vertex number +1 the previous article said. Assuming that the rank of a m*n matrix A is R, the dimension of the left 0 space is m-r.

Here, the dimension of the left 0 space represents the number of unrelated loops. M stands for the number of sides, because 0 of the space in matrix A is 1 dimensions, and the dimension of the column space is r=n-1.

So there is the following formula (i.e. Euler's formula): number of unrelated loops = number of sides-vertices +1

Original:http://blog.csdn.net/tengweitw/article/details/41080571

Nineheadedbird

"Linear algebra" graph and Network