# Random Matrix Algebra

I've been getting familiar with the Sage environment. Here's a nontrivial result (but I dunno how interesting it is) about eigenvalues.

Take the finite field GF(3), and look at all its 3x3 matrices. How many of these 39 = 19683 matrices' characteristic functions have all three roots? Or no roots? Brute force search reveals:

```roots 	count
() 	3456
(0) 	2106
(1) 	2106
(2) 	2106
(0,0,0) 	729
(0,0,1) 	1054
(0,0,2) 	1053
(0,1,1) 	1052
(0,1,2) 	1404
(0,2,2) 	1053
(1,1,1) 	729
(1,1,2) 	1053
(1,2,2) 	1053
(2,2,2) 	729
```

No cubic in GF(3) has just two roots for the same reason as cubics in the reals don't have just two real roots, I think. The element of a field extension in which those two extra roots exist must cancel out. But in GF(3) you've got degree-three extensions too, so zero-root polynomials exist as well.

What's potentially interesting here is the symmetry breaking for (0, 0, 1) and (0, 0, 2). Those cases have +1 and -1 count, respectively, than you would "expect" by comparison with (0, 0, 1), (1, 1, 2), (0, 1, 1), and (1, 2, 2). The single-root cases are perfect symmetric, by contrast.

Anyway, since these roots are the eigenvalues it turns out that just slightly over 50% of the 3x3 matrices in GF(3) have the full set of eigenvalues (and thus might be diagonalizable, although probably a lot are defective.)

Expanding to 4x4 matrices (43 million of them) or GF(5) (about 2 million) would require a lot more processing time, so I'd like to make the computation smarter in some way.
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