We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011.

‎in this paper, we formulate the fourth order sturm-liouville problem (fslp) as a lie group matrix differential equation. by solving this ma- trix differential equation by lie group magnus expansion, we compute the eigenvalues of the fslp. the magnus expansion is an infinite series of multiple integrals of lie brackets. the approximation is, in fact, the truncation of magnus expansion and a gauss...

The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11-13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123-162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39-83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127-172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1-24]. The hypergeometric function kno...

We classify coproducts on matrix algebra in terms of solutions to some modification of pentagon equation. The construction of Baaj and Skan-dalis describing finite dimensional unitary solutions of pentagon equation is extended to the non-unitary case. We establish the relation between Hopf-Galois algebras and solutions to modified pentagon equation.

Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of $2times 2$ matrices.