Norms of Dual Complex Vectors and Dual Complex Matrices

In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Holder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.