Symbolic Computation Group

In this talk, we study the unitary diagonalization of matrices over formal power series rings. We characterize when a normal matrix is unitarily diagonalizable, showing that this happens precisely when its minimal polynomial splits completely over the ring and the corresponding spectral projections remain inside the ring. Based on this characterization, we develop an algorithm for deciding unitary diagonalizability over regular local rings arising from algebraic varieties. A key ingredient is a criterion for determining when a polynomial splits over a formal power series ring, obtained using techniques from prime decomposition and ramification theory.

This is a joint work with Zihao Dai, Hao Liang, and Lihong Zhi.