### M Theory Lesson 252

In vector space arithmetic, instead of ordinary products and additions we have something like tensor products and direct sums. For example, the combination of circulant primes in a matrix ordinal $n$ is a product.

In a paper on random permutation matrices Hambly et al remind us of the observation of Wieand, that the spectrum of such a matrix, although different, has some surprising similarities to that of a random unitary matrix. But the basic Weyl circulant is only one of the MUB factors, coming from the Fourier transform. Combinations of braided (MUB) 1-circulants, with entries that are roots of unity, give rise to more general large complex matrices with one entry in each row and column. Since the eigenvalues of MUB circulants are also always on the unit circle (they are roots of unity) it would be interesting to study the behaviour of the eigenvalues of their products.

Aside: Carl's new Koide paper has many interesting things to say about circulants.

In a paper on random permutation matrices Hambly et al remind us of the observation of Wieand, that the spectrum of such a matrix, although different, has some surprising similarities to that of a random unitary matrix. But the basic Weyl circulant is only one of the MUB factors, coming from the Fourier transform. Combinations of braided (MUB) 1-circulants, with entries that are roots of unity, give rise to more general large complex matrices with one entry in each row and column. Since the eigenvalues of MUB circulants are also always on the unit circle (they are roots of unity) it would be interesting to study the behaviour of the eigenvalues of their products.

Aside: Carl's new Koide paper has many interesting things to say about circulants.

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