Flat portions on the boundary of the indefinite numerical range of \(3\times 3\) matrices (Q924360)

Let \(A\) be an \(n\times n\) complex matrix and \(J= \text{diag}(1,\dots,1,-1,\dots,-1)\) with \(r\) ones and \(n-r\) negative ones. We define the \(J\)-numerical range \(W_{J}(A)\) of \(A\) to be the \(\mathbb R\)-convex subset of \(\mathbb C\) consisting of all values of \(\xi^*JA\xi/\xi^*J\xi\) for \(\xi\in\mathbb C^n\) for which \(\xi^*J\xi\neq0\). A \(J\)-unitary matrix \(U\) is an \(n\times n\) matrix such that \(U^{-1}=JU^*J\), and \(A\) is called \(J\)-unitarily reducible if there exists a \(J\)-unitary matrix \(U\) such that \(A\) and \(J\) can be simultaneously reduced to (proper) block diagonal form via similarity under \(U\). The authors study how the form of the boundary of \(W_{J}(A)\) may determine the kinds of matrices which are \(J\)-unitarily similar to \(A\) in the case \(n=3\). A typical theorem is the following. Let \(J= \text{diag}(1,1,-1)\) and suppose that \(A\) is not \(J\)-unitarily reducible. Then under \(J\)-unitary similarity, translation, rotation and scaling, \(A\) may be written in the form \[ \left[ \begin{matrix} i & 0 & c_{1}\\ 0 & 0 & c_{2}\\ c_{1} & c_{2} & \psi \end{matrix} \right] \] where \(c_{1},c_{2}\) are positive real numbers and \(\operatorname {Re}\psi<0\), if and only if \(W_{J}(A)\) has a closed line segment on its boundary.