Functions of matrices

Abstract

We use the Cayley-Hamilton theorem and the sequence of Horner polynomials associated with a polynomial w(z) to obtain explicit formulas for functions of the form f (tA), where f is defined by a convergent power series and A is a square matrix. We use a well-known explicit formula for the resolvent of A and show that f (tA) is the Hadamard product of f (t) and the function (I − tA) −1 , which is easily obtained from the resolvent.

Fibonacci-Horner decomposition of the matrix exponential and the fundamental system of solutions

“…References [5] and [14] were also pointed out by the referee. We thank Professor L. Verde-Star for his kindness in sending us some of his papers.…”

“…In Verde-Star's approach Horner polynomials play an important role. Moreover, this approach is also considered in [14] for the study of functions of matrices in connection with partial fraction decompositions (see [5]). We shall discuss this aspect in Subsection 4.2 below.…”

“…Second, in [14] Verde-Star has arrived at the result of Theorem 4.2 using the sequence of Horner polynomials {w k (z)} k associated with a given polynomial w(z)…”

“…By the way, we are going to digress from our aim in order to mention that this incorporation allows us to obtain another explicit expression for the sequence {u n } n≥0 in terms of λ j (1 ≤ j ≤ p), which will be useful in improving some results of [1]. Meanwhile, 2) also appears in [14]. We would like to note that it is an interesting issue to illustrate the deep relationship between our development and that of Verde-Star, even though the two approaches are based on different techniques.…”

“…There is a large variety of methods for computing the exponential of a matrix and the differential equation method is among them; see [4,6,7,9,12,13] and references therein. As a consequence, diverse decompositions of e tA have been considered in the literature, associated with various kinds of bases for the vector space of polynomials; see, e.g., [4,6,7,9,12,13,14]. An approach for the decomposition of e tA related to the combinatorial expression of generalized Fibonacci sequences (see [8,10]) is studied in the recent papers [1,2,11].…”