Linear Algebra Done Right · Problem 8A.25
Operators on Complex Vector Spaces — Sheldon Axler
Mathematics — Linear Algebra · nilpotent-operators-orthonormal
Problem
Suppose that is an inner product space and is nilpotent. Show that there is an orthonormal basis of with respect to which the matrix of has the upper-triangular form promised by 8.18(c).Intuitive explanation
The key insight is that nilpotent operators are automatically normal when restricted to invariant subspaces, which lets you apply the spectral theorem machinery. Since is nilpotent, all its eigenvalues are zero, so the upper-triangular form collapses to strictly upper-triangular. Build your orthonormal basis inductively: find an eigenvector (which exists because is nilpotent), orthonormalize it, then work on the orthogonal complement where still acts nilpotently, repeating until done.
Intuitive explanation
The key insight is that nilpotent operators are automatically normal when restricted to invariant subspaces, which lets you apply the spectral theorem machinery. Since is nilpotent, all its eigenvalues are zero, so the upper-triangular form collapses to strictly upper-triangular. Build your orthonormal basis inductively: find an eigenvector (which exists because is nilpotent), orthonormalize it, then work on the orthogonal complement where still acts nilpotently, repeating until done.