Related papers: Explicit matrix inverses for lower triangular matr…
This contribution is motivated by old and recent works on matrix powers and their applications on combinatorial sequences. We give in this paper the $s$-th powers and the inverses for special upper triangular matrices and the $s$-th powers…
A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\chi$-spherical functions indexed by the integers for each Hermitian quantum symmetric pair. A distinguished family of…
Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking…
This paper analyzes the convergence of fixed-point iterations of the form u = f(u) and the properties of the inverse of the related pentadiagonal matrices, associated with the fourth-order nonlinear beam equation. This nonlinear problem is…
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated…
Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
The Cayley-Hamilton problem of expressing functions of matrices in terms of only their eigenvalues is well-known to simplify to finding the inverse of the confluent Vandermonde matrix. Here, we give a highly compact formula for the inverse…
We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y…
There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras…
Commutative hypercomplex algebras offer significant advantages over traditional quaternions due to their compatibility with linear algebra techniques and efficient computational implementation, which is crucial for broad applicability. This…
In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix $A$, generated by the first column of the form…
We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…
We introduce two related notions of pattern enforcement in $(0,1)$-matrices: $Q$-forcing and strongly $Q$-forcing, which formalize distinct ways a fixed pattern $Q$ must appear within a larger matrix. A matrix is $Q$-forcing if every…
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…
We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 x 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,4). The 15 matrices with eigenvalue one correspond to a…
We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of…
In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of ${\mathcal B}(\ell^q_w)$, the algebra of all bounded linear operators on…
This overview article gives an elementary approach to continuous q-Hermite polynomials. We stress their relation to Fibonacci, Lucas and Chebyshev polynomials and to some q-analogues of these polynomials.