conjugation by elementary matrices
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(i) Let G = SL2(R). Using conjugation by elementary matrices, show that every matrix A in G except for +-I is conjugate to a matrix having one of the forms
| 0 -1 |
| 1 d |
or
| 0 1 |
|-1 d |.
(ii) Let
A = | x y |
| z w |
be a matrix in G and let t be its trace. Substituting t-x for w, the condition that det A = 1 becomes x(t-x) - yz = 1. If we fix the trace t, then the locus of solutions of this equation is a quadric in x,y,z-space. Describe the quadrics that arise this way, and decompose them into conjugacy classes.
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This soltion uses conjugation by elementary matrices.
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