eigenspaces

On top of eigenspaces comment, the promise was never that any odd Julia code would be fast.

Let's see what happens in each of the cases: 1) In case 1, there are no eigenspaces and no eigenvalues.

In this way, eigenvalues and eigenspaces are unambiguously geometrically defined, and don't require coordinates or matrices.

We can find such a shared eigevector v as M, M* commute implies that M* will preserve the eigenspaces of M.

Now one of the first things you learn in linear algebra is that if two matrixes commute, they share the eigenspaces.

Otherwise there are two different eigenvalues, and their eigenspaces are two different straight lines.

Very cool, but as a layman I was a very confused by the description of eigenspaces and the S1/S2 lines.

A composite system's states are then restricted to the intersection of the +1/-1-eigenspaces of all the permutation operators (more on the +1/-1 thing later).

> As you can see below, eigenspaces attract this sequence S1 and S2 were displayed earlier, but not explained, now this section implies that those lines are the Eigenspaces?

Schrödinger time evolution is linear and unitary, ie once you split off branches that have factors living in different eigenspaces, they can no longer interact.

> once you split off branches that have factors living in different eigenspaces And how exactly does that happen?

C on eigenspaces and diagonal matricies.

In these examples two exist, labeled, S1 and S2." "Eigenspaces show where there is 'stability' from repeated applications of the eigenvector.

If you compute M^∞, the 1/4 and 1/2 "eigenspaces" will disappear, so you're left with the subspace of the eigenvalue 1.

However, the last sentence also says eigenspaces are (despite the "space" in their name) lines, which throws the earlier interpretation into doubt.

Now let's define the "eigenspaces", or "own spaces" of the transformation, corresponding to each eigenvalue.