Reconstruction of the Hermitian matrix by its spectrum and spectra of some number of its perturbations

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

“…. , n. Then, (24) defines two distinct parallel lines for each r. Moreover, for a fixed r, the intersection of the two lines with the circle occurs at two distinct pairs of points, where a single pair consists of a point and its antipode (see Fig. 2).…”

“…Since the spectral data comes from a pentadiagonal matrix, each of the lines from (24) must have an intersection point |ã (n) or −|ã (n) where |ã (n) is the nth column (with only two nonzero entries) of the original matrix (see Fig. 3).…”

“…These are highly degenerate cases where all the lines are determined by a fixed pair of parallel lines (l + , l − ). More precisely, for every r, the lines from (24) are either equal to (l + , l − ) or to another fixed pair of lines (l + , l − ) where l +/− ⊥ l +/− (see Fig. 4).…”

“…Then, the slope of l +/− is − 1 α . Thus, for every r the slopes of the lines from (24) are either α or − 1 α , i.e., either…”

“…In particular, it has been shown that there exists a finite number of square N × N matrices with a prescribed spectrum and prescribed off-diagonal entries [21]. A different approach to the problem is presented in [24] where any Hermitian matrix can be reconstructed from its spectrum and the spectra of a suitable number of its perturbations. Recently, there has been some revived interest in the so-called eigenvector-eigenvalue identity [10] which allows one to find the amplitudes of eigenvector's entries of a hermitian matrix using the spectra of its (N − 1) × (N − 1)-minors and the spectrum of the full matrix itself.…”