torus is a surface having genus one
is there a possibility to “transform” a matrix (inheriting the input-output “weights”) into a graph object inheriting the matrix relations?
you can convert it to an edge-weighted graph using
WeightedAdjacencyGraph. This will give you a complete graph (a
Graph expression) in which each vertex is also connected to itself. I am not sure how much sense it makes treat this matrix as a graph, given the full connectivity.
Since you have a fully connected graph, it does not matter in your case, but be aware that
WeightedAdjacencyGraph represents missing edges with
Infinity, not with zero. This is somewhat annoying because it is inconsistent with
WeightedAdjacencyMatrix, which uses
Infinity does make sense in some (not all) applications, it is much more common to see data that uses 0.
If V=R2 and W=R2, then T:R2R2 is a linear transformation if and only if there exists a 22 matrix A such that T(v)=Av for all vR2. Matrix A is called the standard matrix for T. The columns of A are T01 and T10 , respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parollelograms to parallelograms. We will often illustrate the action of a linear transformation T:R2R2 by looking at the image of a unit square under T
the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.Affine transformations do not respect lengths or angles; they multiply area by a constant factor
In some cases, however, you may prefer to convert the system of linear equations into a matrix equation, and then apply matrix manipulation operations to solve it. This approach is often useful when the system of equations arises as part of a general algorithm, and you do not know in advance how many variables will be involved.