<i>M</i>th Power of an <i>N × N</i> Matrix and Its Connection with the Generalized Lucas Polynomials

Barakat, Richard; Baumann, Elizabeth

doi:10.1063/1.1664992

Cited by 18 publications

(4 citation statements)

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“…The solution (2) is easily obtained by noticing that Proof. The first formula in (7) follows from direct substitution of D(t) into the given equation (1). The relations (6) are verified by induction [11], The second summation in (7) is then immediate from (6).…”

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confidence: 99%

Dynamic solutions of linear matrix differential equations

Ruiz-Claeyssen¹,

Tsukazan²

1990

Quart. Appl. Math.

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Abstract. We discuss the mth-order linear differential equation with matrix coefficients in terms of a particular matrix solution that enjoys properties similar to the exponential of first-order equations. A new formula for the exponential matrix is established with dynamical solutions related to the generalized Lucas polynomials.1. Introduction. The study of higher-order linear differential equations with square matrix coefficients is usually left aside for being equivalent to a first-order equation having the companion matrix as its coefficient. The handling of this matrix is not an amenable task for obtaining results proper of higher-order equations which are shadowed or simply unknown. The companion matrix, although sparse, does not preserve any common property that the involved matrix coefficients might share.In this paper, we shall present a direct study of higher-order equations in terms of a particular square matrix solution, which is characterized by zero initial "displacement" values and a unit initial "impulse" value, and that we shall refer to in the sequel as the dynamical solution. It will be shown that this solution enjoys intrinsic properties not necessarily shared by the complementary basis solutions, and that in a certain way the dynamical solution should do for higher-order equations what the

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Dynamic solutions of linear matrix differential equations

Ruiz-Claeyssen¹,

Tsukazan²

1990

Quart. Appl. Math.

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“…., r, and therefore of the generalized Lucas polynomials (in r variables) k-1 E Fg,n+r-g-1 (I1,''' Ir)Z n= Various papers have been devoted to the study of the above polynomials (see [21]) and to the extension of the algebraic theory of the Lucas numerical functions (see, e.g., [133, [22]). 1 The functions Fl.n(I1,'", L), n--1, are called in the literature generalized Lucas polynomials (see [2], [21]). The above results have been extended (see [4]) to a matrix whose minimal polynomial is known.…”

Section: An Explicit Formula For F() and The Generating Functions Of mentioning

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An Explicit Formula for $f(\mathcal{A})$ and the Generating Functions of the Generalized Lucas Polynomials

Bruschi¹,

Ricci²

1982

SIAM J. Math. Anal.

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“…(3) is to note that Hn(n;;' 3) can be written as a linear combination of H2, H, and I by virtue of the Cayley-Hamilton theorem. The coefficients of these matrices are polynominals in the three eigenvalues of H. Barakat and Baumann (1969) have shown that these polynomials are expressible in terms of generalized Lucas polynomials, a class of functions they introduced to study the Mth power of an N x N matrix. They show that…”

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