Analytic and Structural Insights into Eigendecompositions of J2 Binormal Unitary Matrices on the Complex Plane

This research focuses on the eigende compositions of J2 binormal unitary matrices, denoted as J_2^bum (r). These matrices encapsulate matrix components with cohesive logical properties and secondary binormal behavior along a specified unit loop r∈J_2. Addressing gaps in current knowledge, we establish the existence of a decomposition J_2^bum (r)=Un(r)Bi(r)Un(r)^P, meeting conditions such that Un(r)^P=Un(r)^* (conjugate transpose), Bi(r) is a real binormal matrix, and both Un(r) and Bi(r) are analytic functions of a positive integer N. The decomposition extends the well-known Rellich theorem to matrix-valued functions exhibiting J2 binormal unitarity on the real number line. Furthermore, this theorem can be adapted for functions with analyticity and J2 binormal unitary characteristics along lines or circles in the complex plane, and the findings are extended to J2 binormal unitary matrices with elements satisfying a unitary series.