Diagonal imbeddings in a normal matrix (Q2444301)

Let \(A\in\mathbb{C}^{n\times n}\), \(B\in\mathbb{C}^{m\times m}\) with \(m\leq n\) and let \(\sigma(A)\) be the spectrum of \(A\). We say that \(B\) is imbeddable in \(A\) if there exists \(W\in\mathbb{C}^{n\times m}\) with orthonormal columns such that \(W^{\ast}AW=B\). In the case \(m=1\), the imbeddable matrices are precisely the \(1\times1\) matrices with entries in the numerical range \(w(A)\) of \(A\). The current paper studies a general inverse numerical range problem for normal matrices: given a normal matrix \(A\) and a point \(\mu_{1}\in w(A)\setminus \sigma(A)\), find the largest integer \(m\geq1\) and \(\mu_{2},\dots,\mu_{m}\in w(A)\setminus\sigma(A)\) such that \(\mathrm{diag}(\mu_{1},\dots,\mu_{m})\) is imbeddable in \(A\). This problem is solved for \(n=3\) and for \(n\geq4\) it is proved that \(m\geq n/3\) (Theorem 1). A number of examples is included.