Diagonalizability Conditions for a (2,1)-Band Matrix of order 4 with a Prescribed Spectrum

Introduction: This work studies the inverse eigenvalue problem for a (2,1)-band matrix of order four. The analysis demonstrates that the transformation matrix is nonsingular if and only if the eigenvalues are pairwise distinct. We ask ourselves what is the minimum number of entries that we can nullify so that it is diagonalizable with an eigenvalue of algebraic multiplicity two.

Methods: Using diagonal similarity transformations, the original matrix family is reduced to a normalized form with unit superdiagonal entries. This normalization preserves the spectrum while reducing the number of effective parameters. We study the characteristic polynomial, and the coefficient comparison yields a nonlinear algebraic system involving the matrix coefficients.

Results: If one eigenvalue has algebraic multiplicity two, the matrix cannot remain diagonalizable while preserving all nonzero band entries. Motivated by this obstruction, we investigated the minimum number of entries that must be nullified in order to admit a diagonalizable matrix with a repeated eigenvalue. For four eigenvalues, one of them double and the others single, the symbolic construction produces a nonsingular transformation matrix T, which establishes the existence of a complete basis of eigenvalues.

Conclusions: We have studied the minimum number of entries that must be set to zero in order to recover diagonalizability in the presence of a double eigenvalue. Explicit symbolic constructions were obtained. In particular, a parametric family of matrices satisfying the prescribed spectral conditions was derived together with a closed-form diagonalizing transformation matrix.