Nilpotency

Definition

A square matrix $A$ is nilpotent iff there exists an integer $k>1$ $s$.$t$. $A^k=0$. The smallest such $k$ is called the index of nilpotency.

Equivalent Characterizations

• All eigenvalues are $0$ (over $\mathbb{C}$ or over $\mathbb{R}$)
• The characteristic polynomial is $P_A(\lambda )={\lambda }^n$
• $A$ is similar to a direct sum of Jordan blocks with eigenvalues $0$

Useful fact

if $A$ is $n\times n$ nilpotent, then:

• $A^n=0$
• $Tr(A^m)=0$ for all $m\ge 1$