Abstract
The spectrum of a real and symmetric Ntimes N matrix determines the matrix up to unitary equivalence. More spectral data is needed together with some sign indicators to remove the unitary ambiguities. In the first part of this work, we specify the spectral and sign information required for a unique reconstruction of general matrices. More specifically, the spectral information consists of the spectra of the N nested main minors of the original matrix of the sizes 1,2,ldots ,N. However, due to the complicated nature of the required sign data, improvements are needed in order to make the reconstruction procedure feasible. With this in mind, the second part is restricted to banded matrices where the amount of spectral data exceeds the number of the unknown matrix entries. It is shown that one can take advantage of this redundancy to guarantee unique reconstruction of generic matrices; in other words, this subset of matrices is open, dense and of full measure in the set of real, symmetric and banded matrices. It is shown that one can optimize the ratio between redundancy and genericity by using the freedom of choice of the spectral information input. We demonstrate our constructions in detail for pentadiagonal matrices.
Highlights
The answer to the question posed in the title is definitely: No! The spectrum determines the matrix up to unitary equivalence
Given that the spectra σ(n), n = 1, . . . , N correspond to a D-diagonal real symmetric matrix, we provide a finite algorithm which produces the set of possible D-diagonal matrices whose main minors have the given spectra
A matrix is determined by its spectrum only up to unitary equivalence
Summary
The answer to the question posed in the title is definitely: No! The spectrum determines the matrix up to unitary equivalence. The answer to the question posed in the title is definitely: No! The spectrum determines the matrix up to unitary equivalence. The question is what additional spectral information is required for this purpose. This question accompanies Mathematical Physics already for a long time, and it is usually referred to as the spectral inversion or spectral reconstruction problems [1,2]. The related reconstruction methods have a wide scope of applications in different areas of Physics, Materials Science and Engineering. They appear, e.g., in studies of mechanical systems near equilibrium, where one tries to construct a
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