Let \(A\) be an \(n\times n\) real matrix such that \((A-I)^m=0\) for some \(m\ge1\). Prove that there exists an \(n\times n\) real matrix \(B\) such that \(B^2=A\).
Solution
Write \(A = I + N\) , so \(N ^ { m } = 0\) . Let \(P ( x )\) be the m-th Taylor polynomial of the function \(\sqrt { 1 + x }\) , so \(P ( x ) ^ { 2 } \equiv 1 + x\) (mod \(x ^ { m } )\) . In other words
\[ P ( x ) ^ { 2 } = 1 + x + x ^ { m } Q ( x ) \]
for some \(Q ( x ) \in \mathbb { R } [ x ]\) . Then
\[ P ( N ) ^ { 2 } = I + N + N ^ { m } Q ( N ) = I + N = A , \]
so \(B : = P ( N )\) satisfies \(B ^ { 2 } = A\) .