Kunze Linear Algebra · Problem 7.2.10

Rational and Jordan Forms

Mathematics — Methods · cyclic-decomposition

Problem

Let FF be a subfield of the complex numbers and let TT be the linear operator on F4F^4 which is represented in the standard ordered basis by the matrix [200012000a2000b2].\begin{bmatrix} 2 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & a & 2 & 0 \\ 0 & 0 & b & 2 \end{bmatrix}. Find the characteristic polynomial for TT. Consider the cases a=b=1a = b = 1; a=b=0a = b = 0; a=0a = 0, b=1b = 1. In each of these cases, find the minimal polynomial for TT and non-zero vectors α1,…,αr\alpha_1, \ldots, \alpha_r which satisfy the conditions of Theorem 3.

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