Orthogonality

Graphs

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:

Gram-Schmidt algorithm