On the Minimum Polynomial and Applications

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

Advances in the Approximation of the Matrix Hyperbolic Tangent

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

Matrix-valued Bratu equation and the exact solution of its initial value problem

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.