Orthogonality
Last updated
Last updated
Graphs, not in the sens of calculus and lines. 1 example: if 1 person is a node and the edges is friendship. Usually the max distance is 6 degrees of separation. These are applications of linear algebra.
Ex:
I need to give a direction to the edges. For example current flowing on the edges, i'll know the direction if its positive of negative.
The incidence matrix for this graph is:
Let's stop at row 3 for a second because edges 1 2 and 3 form a loop. They correspond to linearly dependent rows. Real matrices from real problems have structure.
Let's ask regular matrices questions.
Nullspace? Are the 4 cols independent or dependent. Let's solve Ax=0.
So we've created a matrix that computes the differences across every edge. The differences in potential. In electricity, potential differences is what makes current flow, it's what makes things happen.
The null space is:
dim N(A) = 1. The rank is n - r = 3. Any 3 potential are independent, and typically we ground that node, we make it 0.
The null spaces means that the potentials can only be determined up to a constant.
What is the nullspace of A T? because A T y = 0 is probably the most fundamental equation in applied math.
Let's find a basis for this nullspace.
Now we can find a C that gives us the relation with potentials and the current: that's actually Ohm's law.
So back to A transpose. The equation from A becomes: (and we can see why on the graph).
What is a basis: We know we'll have 2 vectors because dim = 2. Any current around a loop satisfies the current law.
A graph without a loop is called tree.
We have Euler's formula: