We introduce a novel class of self-similar matrices exhibiting recursive structural patterns across multiple scales. These matrices provide a unifying framework that connects discrete linear structures with geometric properties. As a first application, we show that a suitable member of this family defines the Lagrangian Grassmannian of a symplectic vector space of dimension 2n, providing a new constructive characterization of isotropic and Lagrangian subspaces. This approach reveals an intrinsic self-similar organization underlying symplectic geometry. As a second application, we investigate the role of these matrices in coding theory. We show that their recursive structure naturally gives rise to sparse parity-check matrices, leading to families of low-density parity-check (LDPC) codes with highly structured connectivity and efficient algebraic descriptions. The interplay between self-similarity, symplectic geometry, and coding theory opens new directions for both theoretical research and the design of structured error-correcting codes.
This paper provides a complete characterization of the real matrices A for which the equation p(X) = A has a real solution X∈Mn(R) $X\in {M}_{n}\left(\mathbb{R}\right)$ for every nonconstant polynomial p∈R[x] $p\in \mathbb{R}\left[x\right]$ . We prove that this property holds if and only if every real eigenvalue of A is semisimple and has even algebraic multiplicity. Moreover, for a matrix A satisfying these equivalent conditions and a fixed nonconstant polynomial p∈R[x] $p\in \mathbb{R}\left[x\right]$ , we establish a simple spectral criterion for the existence of a real primary solution: such a solution exists if and only if σ(A)∩R⊆p(R) $\sigma \left(A\right)\cap \mathbb{R}\subseteq p\left(\mathbb{R}\right)$ . Consequently, p(X) = A admits a real primary solution for every nonconstant polynomial p∈R[x] $p\in \mathbb{R}\left[x\right]$ if and only if A has no real eigenvalues.
The geometric mean cone (GMC) is a higher-order generalization of the second-order cone and the rotated quadratic cone. While these lower-order cones admit well-established Euclidean Jordan algebraic representations, no analogous framework is currently known for GMCs of order three or higher. In this paper, we study the algebraic structure of the GMC by extending the algebra of the rotated quadratic cone to higher orders. Specifically, we introduce and analyze notions such as a bilinear product, linear matrix representation, characteristic polynomial, rank, and a trace inner product. By adopting the trace inner product, we show that, contrary to certain claims in the existing literature, the GMC is self-dual. While the algebras of both the rotated quadratic cone and the second-order cone are Euclidean Jordan algebras, we show that the extended algebra of the GMC is a Euclidean metrized algebra. However, for orders three and higher, no quadratic representation exists in the extended algebra that maps the entire ambient space into the GMC. Consequently, it remains an open question whether the power cone admits a quadratic representation. Our results clarify the precise sense in which the GMC admits an algebraic structure, with potential implications for logarithmic barrier constructions in interior-point methods over GMCs.
In operator and matrix theory, obtaining sharper bounds for quantities such as the operator norm and the numerical radius is of central importance. The main goal of this paper is to present new novel bounds that work for the norm and numerical radius of accretive–dissipative operator matrices, with particular emphasis on the interaction between diagonal and off-diagonal blocks. The obtained results are compared with the existing literature, showing that the new bounds provide better estimates than some known results.
When does a square matrix have a square root? When does it have a log? The answers to these questions lie in its Jordan form.
