The Inverse Eigenvector Problem for Real Tridiagonal Matrices

A little known property of a pair of eigenvectors (column and row) of a real tridiagonal matrix is presented. With its help we can define necessary and sufficient conditions for the unique real tridiagonal matrix for which an approximate pair of complex eigenvectors are exact. Similarly we can designate the unique real tridiagonal matrix for which two approximate real eigenvectors, with different real eigenvalues, are also exact. We close with an illustration that these unique “backward error” matrices are sensitive to small rounding errors in certain partial sums which play a key role in determining the matrices.