What is a nilpotent matrix?
Solution
A square matrix \(A\) over a ring or field is nilpotent if \[ A^r=0 \] for some integer \(r\ge1\).
The least such \(r\), when it exists, is the nilpotency index of \(A\).
Over a field, the following are equivalent:
- \(A\) is nilpotent;
- the minimal polynomial of \(A\) is \(x^r\) for some \(r\ge1\);
- over an algebraic closure, every eigenvalue of \(A\) is \(0\);
- the Jordan form of \(A\) consists entirely of nilpotent Jordan blocks \(J_s(0)\).
In particular, an \(n\times n\) nilpotent matrix satisfies \(A^n=0\).