Math 6310 • Algebra

Let \(A\) be a real \(n\times n\) matrix. We say that \(A\) is a \emph{difference of two squares} if
there exist real \(n\times n\) matrices \(B\) and \(C\) with \(BC=CB=0\) and \(A=B^2-C^2\).
\begin{enumerate}
\item If \(A\) is a diagonal matrix, show that it is a difference of two squares.
\item If \(A\) is a symmetric matrix that is not necessarily diagonal, again show that it is a
difference of two squares.
\item Suppose \(A\) is a difference of two squares, with corresponding matrices \(B\) and \(C\) as
above. If \(B\) has a nonzero real eigenvalue, prove that \(A\) has a positive real eigenvalue.
\end{enumerate}