Kunze Linear Algebra · Problem 7.3.15

Rational and Jordan Forms

Mathematics — Methods · nilpotent-square-root

Problem

N

is a nilpotent

3 \times 3

matrix over

\mathbb{C}

, prove that

A = I + \frac{1}{2}N - \frac{1}{8}N^2

satisfies

A^2 = I + N

, i.e.,

A

is a square root of

I + N

. Use the binomial series for

(1 + t)^{1/2}

to obtain a similar formula for a square root of

I + N

, where

N

is any nilpotent

n \times n

matrix over

\mathbb{C}