The Number of 2 by 2 Matrices over 핫p핫 with Eigenvalues in the Same Field
Gregor Olšavský · Mathematics Magazine · 2003
As an example of the power of Theorem 3, consider the following observations. We remarked above that Mo is both S-odd and S-even for every symmetry S. It is easy to check that Mo is the only function that is both odd and even. Since every conjugate of Mo is Mo, the only function that is S-odd and S-even for any symmetry is Mo. It is also easy to see that no even function can be strictly monotonic in every open interval containing zero. By Theorem 3 this is true for S-even functions with respect to any symmetry. The odd functions x, -x, and x2 sin(l/x) demonstrate that S-odd functions can be monotonically increasing, decreasing, or neither near zero. We might suspect that conjugation gives an easier way to determine the S-odd and S-even parts of a function f. Let h = (M S)-1. Then S = ho S o h = M_1. The conjugate of f, f = h-' o f o h, has a unique decomposition into odd and even parts: f =f + fe. The inverse conjugates fo and fe of fo and fe are S-odd and S-even, respectively. But unless S = M_1 it is not true that f = fo + fe. We conclude with a rather surprising theorem.