### Problem with the Matrix

As many AF readers know, Carl Brannen has been diligently trying to solve the magic matrix decomposition problem for $3 \times 3$ unitary matrices. Now Lubos Motl very helpfully decided to solve this problem for Carl, also providing witty commentary along the lines of

Let me give you some examples because they’re easier for you than mathematics. Math class is hard, Barbie.Unfortunately, he made a very elementary mistake and failed to solve the problem after all, but was somewhat annoyed, it would seem, when Carl pointed this out to him. I am informed that Carl has now been banned from Motl's blog. It is a shame that we cannot continue this fruitful collaboration because, as Carl often points out, Motl has made lasting contributions to physics with his concept of tripled Pauli statistics. We will just have to continue playing with our simple matrices alone.

## 6 Comments:

Dear Lubos, unfortunately I am still fairly impoverished and liquidation could only be a positive thing for me financially. Apologies if I have misunderstood the situation with regards to magic matrices.

I should defend Lubos a little here. It's a matter of the difference between "from a unique" and "from the unique".

He interpreted the sentence "Every unitary 3x3 matrix can be obtained from a unique magic unitary matrix by multiplying rows and columns by arbitrary complex phases." as if it had instead been "the unique magic unitary matrix" and proceeded to show that there was no such unique matrix.

Mathematicians get used to speaking like this but it doesn't necessarily translate to well into physics speak. Along that line, a better way of putting it would be to write "Every unitary 3x3 matrix can be obtained uniquely from a magic unitary matrix", but the "from a unique" construct is not uncommon in mathematics.

The problem with Lubos and most other professionals is that they're not attempting to obtain information in this sort of thing. They're only looking for errors, and if they can misinterpret things, so much the better, they find an error more quickly (theirs). This is why you can't teach an old dog new tricks; their hearing is failing.

By the way, the thought that Lubos would sue you or me for defaming him is multidimensionally laughable. He already has an awful reputation, neither of us has any money, he was mathematically wrong, and in any case we haven't defamed him.

Yes, to be fair, Lubos is not mistaken that I am often quite stupid and truly obsessed with these matrices. But then I do try to forgive myself for this, since I care about the physics. And as an old dog myself, I do hope to see more young people playing with these questions.

From my personal experience, Lubos does occasionally admit a mistake - but only after having made it very embarassing for himself, and only if forced to do it by checking the matter with people who know better.

I think neither of those clauses apply in this case, so I do not really know what you were expecting... His reaction here is quite typical, but I know that nor you nor Carl will take it too personally...

Cheers,

T.

Perhaps Lubos did contribute to the proof; a little infamy improved the advertising and so PhilG stepped up with a nice proof, based on the observation that a row-column phase transformation of a unitary matrix to magic form is equivalent to the much simpler problem of finding two vectors of phases u and v, such that

uUv = m

where U is the unitary matrix, and m is the sum for the magic unitary matrix (which, as it turns out, we might as well assume is equal to 1). However, the non constructive proof does not show that the magic matrix so obtained is unique.

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