DOI: 10.15672/hujms.1898416 ISSN: 2651-477X

Complex Adjoint Representations and Eigenvalue Theory for Cartan Number Matrices

Hasan Çakır, İskender Öztürk, Mustafa Özdemir
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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 $2\times 2$ representation thatestablishes an algebra isomorphism between $\mathbb{S}$ and a Hermitian-typesubalgebra of $M_{2}(\mathbb{C})$.Based on this representation, we define the complex adjoint matrix$\Xi_{\mathbf{K}}$, prove the identity$\Xi_{\mathbf{K}\mathbf{M}}=\Xi_{\mathbf{K}}\Xi_{\mathbf{M}}$, and show that theinvertibility of $\mathbf{K}$ is equivalent to that of $\Xi_{\mathbf{K}}$,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 $\Xi_{\mathbf{K}}$ fully characterize eigenvalues in $\mathbb{C}$.

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