On the Convergence of the Sequence of Powers of a $2 \times 2$ Matrix

by Roman W. Wong

This article originally appeared in:
Mathematics Magazine
June, 1996

Subject classification(s): Linear Algebra | Eigenvalues and Eigenvectors | Statistics and Probability | Probability | Stochastic Processes
Applicable Course(s): 3.8 Linear/Matrix Algebra | 7.3 Stochastic Processes

The fact that the limit of the $n$-th power of a $2\times 2$ matrix $A$ tends to $0$ if  $\det A < 1$ and $\mid 1 + \det(A) \mid > \mid$ tr$(A) \mid$ is used to prove a well-known theorem in Markov chains for $2 \times 2$ regular stochastic matrices and to obtain an explicit formula for the stationary matrix and eigenvector.

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Capsule Course Topic(s):
Linear Algebra | Application: Markov