Kunze Linear Algebra · Problem 7.2.3

Rational and Jordan Forms

Mathematics — Methods · t-invariant-complementary-subspace

Problem

Let TT be the linear operator on R3R^3 which is represented in the standard ordered basis by the matrix [200120003].\begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}. Let WW be the null space of T−2IT - 2I. Prove that WW has no complementary TT-invariant subspace. (Hint: Let β=ϵ1\beta = \epsilon_1 and observe that (T−2I)β(T - 2I)\beta is in WW. Prove there is no α\alpha in WW with (T−2I)β=(T−2I)α(T-2I)\beta = (T-2I)\alpha.)

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