Complex Adjoint Representations and Eigenvalue Theory for Cartan Number Matrices

Cartan numbers form a four-dimensional real algebra with a mixedhyperbolic--nilpotent structure and an indefinite quadratic character, whichprecludes the induced modulus from defining a norm and distinguishes theirmatrix theory from that of classical hypercomplex systems.In this paper, we develop a systematic framework for matrices over the algebraof Cartan numbers by introducing a complex 2× 2 representation thatestablishes an algebra isomorphism between S and a Hermitian-typesubalgebra of M₂(C).Based on this representation, we define the complex adjoint matrixΞK, prove the identityΞKM=ΞKΞM, and show that theinvertibility of K is equivalent to that of ΞK,leading to the definition of a q-determinant with multiplicativity andsimilarity invariance properties.We further investigate right eigenvalues and generalized right eigenvalues ofCartan number matrices and matrix pencils, demonstrating that non-real Cartaneigenvalues generate infinite similarity orbits and that the complex spectraof ΞK fully characterize eigenvalues in C.