Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler
Mathematics — Linear Algebra · upper-triangular-matrices
Problem
Give an example of an operator whose matrix with respect to some basis contains only 0's on the diagonal, but the operator is invertible.
This exercise and the exercise below show that 5.41 fails without the hypothesis that an upper-triangular matrix is under consideration.