Final state of a two-dimensional dynamical system

Calculus Level pending

A two-dimensional dynamical system is characterised by the following state equation:

\[ x(k+1) = A x(k) \]

where

\[ x(k) = \begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix} \]

is the state vector, and matrix \( A \) is given by

\[ A = \begin{bmatrix} 0.8 && 0.3 \\ 0.2 && 0.7 \end{bmatrix} \]

Given that

\[ x(0) = \begin{bmatrix} 1 \\ 4 \end{bmatrix} \]

Then what is,

\[ \lim_{k \to \infty} x(k) ?\]

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