Trace of Positive Integer Power of Real 2 × 2 Matrices

Affiliation(s)

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

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

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* (Tr*A*) and Determinant of *A* (Det*A*), 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 Tr*A*^{n} when Tr*A* or Det*A* are zero.

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

Cite this paper

Pahade, J. and Jha, M. (2015) Trace of Positive Integer Power of Real 2 × 2 Matrices.*Advances in Linear Algebra & Matrix Theory*, **5**, 150-155. doi: 10.4236/alamt.2015.54015.

Pahade, J. and Jha, M. (2015) Trace of Positive Integer Power of Real 2 × 2 Matrices.

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_

[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_