Trace of Positive Integer Power of Real 2 × 2 Matrices

Jagdish Pahade^*, Manoj Jha

Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, India.

DOI: 10.4236/alamt.2015.54015                

Abstract

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 TrAⁿ when TrA or DetA are zero.

Keywords

Trace, Determinant, Matrix Multiplication

Received 21 September 2015; accepted 4 December 2015; published 7 December 2015

1. 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(A³)/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 Aⁿ and A⁻ⁿ, n ∊ Z, is proposed in [4] .

Trace of a matrix is defined to be the sum of the elements on the main diagonal of A, i.e.

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 denote its eigenvalues. For q ∊ R, A^q is also symmetric positive definite, and it holds [7] .

(1.1)

This formula is restricted to the matrix A. Also we have other formulae [8] to compute the trace of matrix power such that

(1.2)

But for many cases, this formula is time consuming. For example

Consider a matrix and let we are to find TrA⁵. Eigenvalues of A are, then by (1.2),.

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 Aⁿ is based on the multiplication of matrix.

2. Main Result

Theorem 1. For even positive integer n and 2 × 2 real matrix A,

Proof. Consider a matrix where are real.

Then

(2.1)

and

(2.2)

Now

Then

(2.3)

Now again

(2.4)

Then

(2.5)

Now replace A by A² in (2.3), we have

(2.6)

Again replace A by A² in (2.5), we have

(2.7)

Now again replace A by A² in (2.6), we have

(2.8)

Now we observe from (2.3), (2.6), (2.7) and (2.8) that

Continuing this process up to n terms we get

(2.9)

Finally from above, we get

(2.10)

Hence the proof is completed.

Theorem 2. For odd positive integer n and 2 × 2 real matrix A,

Proof. Consider a matrix A as in theorem 1, we have from (1.4) and (1.6).

(2.11)

Now we observe from (2.5) and (2.11) that

Now we continuing this as in Theorem 1, we get TrAⁿ same as Theorem 1. But here r varies up to. Hence the theorem follows.

Corollary 1: For any positive integer n and 2 × 2 real singular matrix A,.

Proof: For singular matrix A, DetA = 0. Hence proof follows from Theorem 1 and Theorem 2.

Corollary 2: For 2 × 2 real matrix A with TrA = 0.

1) when n is even and;

2) when n is odd.

Proof. Proof follows from theorem 1 and theorem 2.

Corollary 3: For 2 × 2 real matrix A with TrA = 0 and DetA = 0.

where n is any positive integer.

Proof. Proof follows from Corollary 2.

Example 1. Consider a matrix and let we are to find TrA⁵.

Here and. then by Theorem 2, we have

Example 2. Consider a matrix and let we are to find TrA¹⁰.

Here and. then by Theorem 1, we have

Example 3. Consider a matrix and let we are to find TrA²⁰¹⁵.

Here TrA = 0, DetA = −2 and n = 2015, which is odd, hence by corollary 2, we get TrA²⁰¹⁵ = 0.

Example 4. Consider a matrix and let we are to find TrA¹⁰⁰.

Here A is a singular matrix with Trace 1, and then by Corollary 1, we have

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.

Acknowledgements

We would like to hardly thankful with great attitude to Director, Maulana Azad National Institute of Technology, Bhopal for financial support and we also thankful to HOD, Department of Mathematics of this institute for giving me opportunity to expose my research in scientific world.

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Brezinski, C., Fika, P. and Mitrouli, M. (2012) Estimations of the Trace of Powers of Positive Self-Adjoint Operators by Extrapolation of the Moments. Electronic Transactions on Numerical Analysis, 39, 144-155.
[2]	Avron, H. (2010) Counting Triangles in Large Graphs Using Randomized Matrix Trace Estimation. Proceedings of Kdd-Ldmta’10, 2010.
[3]	Zarelua, A.V. (2008) On Congruences for the Traces of Powers of Some Matrices. Proceedings of the Steklov Institute of Mathematics, 263, 78-98.
[4]	Pan, V. (1990) Estimating the Extremal Eigenvalues of a Symmetric Matrix. Computers & Mathematics with Applications, 20, 17-22.
[5]	Datta, B.N. and Datta, K. (1976) An algorithm for Computing Powers of a Hessenberg Matrix and Its Applications. Linear Algebra and its Applications, 14, 273-284.
[6]	Chu, M.T. (1985) Symbolic Calculation of the Trace of the Power of a Tridiagonal Matrix. Computing, 35, 257-268.
[7]	Higham, N. (2008) Functions of Matrices: Theory and Computation. SIAM, Philadelphia.
[8]	Michiel, H. (2001) Trace of a Square Matrix. Encyclopedia of Mathematics, Springer. https://en.wikipedia.org/wiki/Trace_