Linear Algebra, 5th Edition

5.3* Matrix Limits and Markov Chains

In this section, we apply what we have learned thus far in Chapter 5 to study the limit of a sequence of powers $A,A^2,\dots ,A^n,\dots ,$ where A is a square matrix with complex entries. Such sequences and their limits have practical applications in the natural and social sciences.

We assume familiarity with limits of sequences of real numbers. The limit of a sequence of complex numbers $\{z_m:m=1,2,\dots \}$ can be defined in terms of the limits of the sequences of the real and imaginary parts: If $z_m=r_m+i{s}_m$, where $r_m$ and $s_m$ are real numbers, and i is the imaginary number such that $i^2=-1$, then

$$\underset{m\to \infty }{\lim }{z}_m=\underset{m\to \infty }{\lim }{r}_m+i\underset{m\to \infty }{\lim }{s}_m,$$

provided that $\underset{m\to \infty }{\lim }{r}_m$ and $\underset{m\to \infty }{\lim }{s}_m$ exist.

Definition.

Let L, $A_1,A_2,\dots$ be $n\times p$ matrices having complex ...