Inverse Extremal Eigenvalue Problems for Multi-Arrowhead Pentadiagonal Matrices

We address the inverse extremal eigenvalue problem (IEEP) for multi-arrowhead pentadiagonal matrices, a structured class that combines pentadiagonal bandwidth with an alternating arrowhead structure. We identify four distinct structural classes based on the orientation and configuration of arrowhead blocks. For symmetric matrices, we establish sufficient conditions for reconstruction from extremal (smallest, largest, or both) eigenvalues of leading principal submatrices. For certain classes, we prove these conditions are also necessary, providing a complete characterization. Nonsymmetric matrices require one additional prescribed eigenvector. Our results are constructive and yield algorithmic procedures. Numerical examples illustrate our theoretical findings. This work generalizes inverse extremal eigenvalue theory for arrowhead matrices to these pentadiagonal structures.