Interactive lab · the operation itself
Matrix multiplication is one rule applied four times. Click any answer cell and the page shows you exactly which numbers made it.
A worksheet, not a result. The mathematics on this page is standard topological quantum computation, roughly 1984–2009. None of it is new. “It was simply new to my hands. I had read it for years and never computed with it, and those turn out to be different states of knowing.” These sheets exist to move a reader from the first state to the second, and they make no claim of their own.
The method is predict–then–check. Every input box, drill and gate here is built for one move: write down what you think the number is before you reveal it. Numbers are sourced to a paper where a paper says them, and marked unchecked where none does — a sheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust.
If you are coming from an operator formalism, you may want to ask of every quantity here what the declared observable is, and what mapping puts it into the domain before the algebra acts. That is a fair question, and these sheets sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way it is answered. They take no position on it. The full note is on the index.
Prerequisites: none. Pairs with the Golden Arithmetic lab; either order works.
The Verlinde formula is not matrix multiplication. It is a sum of products of individual entries — you never multiply a matrix by anything. That is why hunting for the matrix-multiply step didn't work: there isn't one.
Matrix multiplication turns up in the machinery around Verlinde — diagonalisation, eigenvalues, repeated application. That's what this page is for. Learn it here, and both pages stop having a hole in them.
To get the entry in row i, column j of the answer: take row i of the left matrix, column j of the right matrix, multiply them together term by term, and add.
That's it. The whole operation is that sentence, repeated once per answer cell.
When a cell is selected, the row it came from lights up in A and the column it came from lights up in B. Every answer cell uses exactly one row and one column. Nothing else in either matrix touches it.
Same two matrices. The answers are blank — fill them in and check.
For ordinary numbers 3 × 5 = 5 × 3. For matrices this usually fails. AB and BA are different operations, because the rows and columns swap roles.
A vector is just a matrix one column wide, so the same rule applies. This is the case that matters for what you will see in tutorial 1: apply the Fibonacci fusion matrix over and over to a starting vector.
| step | vector | arithmetic | ratio v₂/v₁ |
|---|
The components are the Fibonacci numbers, and the ratio walks toward φ = 1.618034. You are watching the Perron–Frobenius eigenvalue appear — no algebra, just the same rule applied eight times.
Most vectors get turned as well as stretched. A few come out pointing exactly the way they went in, only longer or shorter. Those are eigenvectors, and the stretch factor is the eigenvalue: N v = λ v.
Those two eigenvalues — φ and −1/φ — are the same two numbers the Verlinde lab reads off the columns of the S-matrix as Sτx/S0x. That is not a coincidence: the columns of S are the eigenvectors of the fusion matrix. Verlinde's theorem is precisely the statement that one matrix, S, diagonalises them all at once.
So the arithmetic on this page and the arithmetic on that one are the same arithmetic, seen from two sides.