Normal matrices and the completion problem (Q2784386)

Let \(M_n({\mathbf F}) \) be the algebra of \(n\times n\) matrices over a field \({\mathbf F}\), where \({\mathbf F}\) is either the field of real numbers \({\mathbf R}\) or of complex numbers \({\mathbf C}\). An inverse problem for matrices (IPFM) is the problem of existence of a matrix \(A\) from a certain class belonging to a given variety \({\mathcal A} \subset M_n({\mathbf F}) \) of dimension \(m\). IPFM is called tight IPFM (TIPFM) if it is given by \(m\) (the dimension of variety \({\mathcal A}\)) algebraically independent over \({\mathcal A}\) polynomial conditions \(p(A)=\alpha\), \(\alpha \in {\mathbf F}^m\) (\(p: M_n({\mathbf F})\to {\mathbf F}^m \) is a polynomial map). NEWLINENEWLINENEWLINEIn the paper under review a certain TIPFM over \({\mathbf R}\) which deals with normal complex-valued matrices with prescribed upper triangular entries is solved. Let \({\mathcal N}_n \subset M_n({\mathbf C})\) be the real variety of normal matrices. It is known that \({\mathcal N}_n\) is an irreducible variety of real dimension \((n^2+n)\) [see \textit{Kh.~D.~Ikramov}, Comput. Math. Math. Phys. 38, No.~1, 1-6 (1998; Zbl 0949.15020)]. The purpose of this paper is twofold. The first one is to describe the structure of \({\mathcal N}_n\), to describe the manifold of regular (smooth) points \({\mathcal N}_n^r\) of \({\mathcal N}_n\), and to describe its covering space induced by the spectral decomposition of a normal matrix. In particular, the author proves that \({\mathcal N}_n^r\) is connected and is not orientable for \(n\geq 2\). The second one is to show that any upper triangular matrix \(A\in M_n({\mathbf C})\) can be completed to a normal matrix \(X \in M_n({\mathbf C})\). Namely, by using the degree theory it is proved that for any \({n+1 \choose 2}\) complex numbers \(a_{i,j}\), \(1\leq i\leq j \leq n\), there exists \(X=(x_{i,j})^n_1\in {\mathcal N}_n\) such that \(x_{i,j}=a_{i,j}\) for all \(i,j\), \(1\leq i\leq j \leq n\). The cases \(n=2,3\) of the second problem were settled by \textit{Kh.~D.~Ikramov} [Math. Notes 60, No. 6, 649-657 (1996), translation from Mat. Zametki, 60, No.~6, 861-872 (1996; Zbl 0897.15010)]. The completion problem was raised by L.~Elsner.