Trace of Positive Integer Power of Real 2 × 2 Matrices

The purpose of this paper is to discuss the theorems for the trace of any positive integer power of 2 × 2 real matrix. We obtain a new formula to compute trace of any positive integer power of 2 × 2 real matrix A, in the terms of Trace of A (TrA) and Determinant of A (DetA), which are based on definition of trace of matrix and multiplication of the matrixn times, where n is positive integer and this formula gives some corollary for n A Tr when TrA or DetA are zero.


Introduction
Traces of powers of matrices arise in several fields of mathematics, more specifically, Network Analysis, Numbertheory, Dynamical systems, Matrix theory, and Differential equations [1].When analyzing a complex network, an important problem is to compute the total number of triangles of a connected simple graph.This number is equal to Tr(A 3 )/6, where A is the adjacency matrix of the graph [2].Traces of powers of integer matrices are connected with the Euler congruence [3], an important phenomenon in mathematics, stating that ( ) ( )( for all integer matrices A, all primes p, and all r ∊ Z.The invariants of dynamical systems are described in terms of the traces of powers of integer matrices, for example in studying the Lefschetz numbers [3].There are many applications in matrix theory and numerical linear algebra.For example, in order to obtain approximations of the smallest and the largest eigenvalues of a symmetric matrix A, a procedure based on estimates of the trace of A n and A −n , n ∊ Z, is proposed in [4]. is defined to be the sum of the elements on the main diagonal of A, i.e.A  The computation of the trace of matrix powers has received much attention.In [5], an algorithm for computing ( ) is proposed, when A is a lower Hessenberg matrix with a unit codiagonal.In [6], a symbolic calculation of the trace of powers of tridiagonal matrices is presented.Let A be a symmetric positive definite matrix, and let { } k λ denote its eigenvalues.For q ∊ R, A q is also symmetric positive definite, and it holds [7]. Tr This formula is restricted to the matrix A. Also we have other formulae [8] to compute the trace of matrix power such that Computation of this value is time consuming.Therefore, other formulae to compute trace of matrix power are needed.Now we give new theorems and corollaries to compute trace of matrix power.Our estimation for the trace of A n is based on the multiplication of matrix.

Main Result
where , , , a b c d are real.

Then
Tr and 3), we have Again replace A by A 2 in (2.5), we have Continuing this process up to n terms we get (2.9) Finally from above, we get Hence the proof is completed.Theorem 2. For odd positive integer n and 2 × 2 real matrix A,      ( ) Now we continuing this as in Theorem 1, we get TrA n same as Theorem 1.But here r varies up to ( )

Conclusion and Future Work
After to discuss Theorems 1 and 2, Corollaries 1, 2 and 3, we are able to find trace of any integer power of a 2 × 2 real matrix.In future, we can be developed similar results for 3 × 3 real matrices.

Theorem 1 .
For even positive integer n and 2 × 2 real matrix A, Consider a matrix A as in theorem 1, we have from (1.4) and (1.6).

a
abc bc a d a b b c bd a d a bc b a d ac a d bc cd bc a d bcd d c a d bc d abc bc a d c a d a b b c bd a d b a d ac a d bc cd bc d bc a d bcd d

Corollary 1 : 2 : 3 :Example 1 .AExample 4 .A
For any positive integer n and 2 × 2 real singular matrix A, For singular matrix A, DetA = 0. Hence proof follows from Theorem 1 and Theorem 2. Corollary For 2 × 2 real matrix A with TrA = 0odd.Proof.Proof follows from theorem 1 and theorem 2. Corollary For 2 × 2 real matrix A with TrA = 0 and DetA = 0. Tr 0 n = A where n is any positive integer.Proof.Proof follows from Corollary 2. Consider and let we are to find TrA 2015 .Here TrA = 0, DetA = −2 and n = 2015, which is odd, hence by corollary 2, we get TrA 2015 = 0. Consider and let we are to find TrA 100 .Here A is a singular matrix with Trace 1, and then by Corollary 1, we have