Give an example of a \(2 \times 2\) matrix \(A\) such that \(A^2\) is normal, but \(A\) is not normal.
Solution
Goal. Find a \(2\times 2\) matrix \(A\) with \(A^2\) normal but \(A\) not normal.
1. Take \(A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\).
a candidate.
2. \(A^2 = 0\), which is normal.
\(A^2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\), and the zero matrix is normal (\(0^* 0 = 0 = 0\,0^*\)).
3. \(A\) is not normal.
3.1. \(A^* = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\).
conjugate transpose.
3.2. \(A^* A = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\).
multiply.
3.3. \(A A^* = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\).
multiply.
3.4. \(A^* A \neq A A^*\), so \(A\) is not normal.
\(\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \neq \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\).
4. Q.E.D.
2 and 3 show \(A^2\) is normal but \(A\) is not.