A complete solution of the normal Hankel problem
The object of the paper is to characterize the \(n\times n\) Hankel matrices which are normal. If \(P_{n}\) is the \(n\times n\) matrix whose \((i,j)\)-th entry is \(1\) if \(i+j=n+1\) and \(0\) otherwise (the ``backward identity'' matrix) and \(H\) is Hankel then \(T:=HP_{n}\) is Toeplitz. It is easily seen that \(H\) is normal if and only if \(TT^{\ast}\) is real. Equivalently, writing \(T=T_{1}+iT_{2}\) where \(T_{1}\) and \(T_{2}\) are real, the Hankel matrix \(H\;\)is normal if and only if \(T_{1}T_{2}^{t} =T_{2}T_{1}^{t}\). Using a case-by-case analysis the authors determine the \(T\) for which \(H=TP_{n}\) is normal Hankel.
scientific article; zbMATH DE number 5706978