Linear Algebra Done Right · Problem 5D.12

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose T∈L(C3)T \in \mathcal{L}(\mathbf{C}^3) is such that 66 and 77 are eigenvalues of TT. Furthermore, suppose TT does not have a diagonal matrix with respect to any basis of C3\mathbf{C}^3. Prove that there exists (z1,z2,z3)∈C3(z_1, z_2, z_3) \in \mathbf{C}^3 such that T(z1,z2,z3)=(6+8z1,7+8z2,13+8z3).T(z_1, z_2, z_3) = (6 + 8z_1, 7 + 8z_2, 13 + 8z_3).

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