On the relations between antitrace, row/column sum of elements and generalized horadam sequence associated with a matrix

Abstract

Using Cayley-Hamilton theorem for a given matrix A = [aij] ∈ M2(K), (K = ℝ or ℂ),  we obtain the recurrence relations for Tra(An), suRi(An) and suCi(An);  Tra(An+2) = Tr(A)Tra(An+1) – det(A) Tra(An) suRi(An+2) = Tr(A)suRi(An+1) – det(A)suRi(An) suCi(An+2) = Tr(A)suCi(An+1) – det(A) suCi(An)    where Tra(A) = a12 + a21  is the sum of antidiagonal entries of A and suRi(A) or scCu(A) is the ith(i = 1, 2) row or column sum of A respectively. In doing so, the analogy between the above natural recurrences and the Horadam sequence is observed. Motivated by this, we define the Horadam sequence {Wn(a, b : Tr(A), det(A))} and Fibonacci sequence {Fn = Wn (0, 1; Tr(A), det(A))} associated with A. In this paper, Tra(An), {suRi(An)}, {suCi(An)}, {Wn} and {Fn} sequences are defined and their inter-relationships are extensively studied. We also obtain Binet’s formulae, generating functions, Cassini identities, Vajda’s identities and Catalan’s identities for these sequences. Further, identities are illustrated for particular values of trace and determinant of a matrix