Consider a stochastic matrix, P and initial condition x0. To find the longterm b

Question

Consider a stochastic matrix, P and initial condition x0. To find the longterm behavior of a system two students perform these computations: Anna computes: P^1000 x0 and says that this represents the behavior of the system at t = 1000. Berna computes the dominant eigenvalue and dominant eigenvector. Berna says that the dominant eigenvalue (normalized) describes the long term behavior of the system. Only Anna's method is correct Only Berna's method is correct Both methods are correct Neither method is correct

Explanation / Answer

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize transitions for a finite Markov chain, Elements of the matrix must be real numbers in the closed interval [0, 1].

Eigenvalues & Eigenvectors

In linear algebra we learned that a scalar l is an eigenvalue for a square n´n matrix A if there is a non-zero vector w such that Aw = lw, we call the vector w an eigenvector for matrix A. The eigenvalue acts like scalar multiplication instead of matrix multiplication for the vector w. Eigenvalues are important for many applications in mathematics, physics, engineering and other disciplines.

The Dominant Eigenvalue

A n´n matrix A will have n eigenvalues (some may be repeated). By the dominant eigenvalue we refer to the one that is biggest in terms of absolute value. This would include any eigenvalues that are complex.

both moethods are correct

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