Construction
of a Solution to Horn’s Conjecture via Inverse Problem of Band Matrices
Yilin Yang, Drexel University, Class of 2017
Hermitian matrices have a wide range of applications in physics and engineering, and are closely and non-trivially linked with many other mathematical objects. A. Horn's problem is a long-standing mathematics problem that asks to characterize eigenvalues of Hermitian matrices A, B and A+B. Although it was solved in the beginning of this century, the proof requires a heavy machine outside of linear algebra in which the original problem stands and the construction of A and B with valid spectral data remains unclear. In this research, we make a connection between A. Horn's problem and another classical linear algebra problem, inverse problem of band matrices, via the rank of A and bandwidth of B. This connection gives an efficient way to build a solution for A. Horn's problem with given spectral data. In the base case where A has rank 1, we show how the Hermitian matrices A, B and A+B with given sets of eigenvalues are constructed through recovering B from the eigenvalues of B and the eigenvalues of the submatrix obtained by removing the first row and column of B.
Yilin Yang, Drexel University, Class of 2017
Hermitian matrices have a wide range of applications in physics and engineering, and are closely and non-trivially linked with many other mathematical objects. A. Horn's problem is a long-standing mathematics problem that asks to characterize eigenvalues of Hermitian matrices A, B and A+B. Although it was solved in the beginning of this century, the proof requires a heavy machine outside of linear algebra in which the original problem stands and the construction of A and B with valid spectral data remains unclear. In this research, we make a connection between A. Horn's problem and another classical linear algebra problem, inverse problem of band matrices, via the rank of A and bandwidth of B. This connection gives an efficient way to build a solution for A. Horn's problem with given spectral data. In the base case where A has rank 1, we show how the Hermitian matrices A, B and A+B with given sets of eigenvalues are constructed through recovering B from the eigenvalues of B and the eigenvalues of the submatrix obtained by removing the first row and column of B.