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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
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