Can One Hear a Matrix? Recovering a Real Symmetric Matrix from Its Spectral Data

Abstract The spectrum of a real and symmetric $$N\times N$$ N × matrix determines the up to unitary equivalence. More spectral data is needed together with some sign indicators remove ambiguities. In first part this work, we specify information required for unique reconstruction general matrices. specifically, consists spectra N nested main minors original sizes $$1,2,\ldots ,N$$ 1 , 2 … . However, due complicated nature data, improvements are in order make procedure feasible. With mind, second restricted banded matrices where amount exceeds number unknown entries. It shown that one can take advantage redundancy guarantee generic matrices; other words, subset open, dense full measure set real, optimize ratio between genericity by using freedom choice input. We demonstrate our constructions detail pentadiagonal