Find all fixed points for the iterative systems with the...

Find all fixed points for the iterative systems with the following coefficient matrices: (a) $\left(\begin{array}{ll}7 & .3 \\ .2 & .8\end{array}\right)$, (b) $\left(\begin{array}{rr}.6 & 1.0 \\ .3 & -.7\end{array}\right)$, (c) $\left(\begin{array}{rrr}-1 & -1 & -4 \\ -2 & 0 & -4 \\ 1 & -1 & 0\end{array}\right)$, (d) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 2 & 3 & -2 \\ -1 & -1 & 2\end{array}\right)$.

Find all fixed points for the iterative systems with the following coefficient matrices:
(a) $\left(\begin{array}{ll}7 & .3 \\ .2 & .8\end{array}\right)$,
(b) $\left(\begin{array}{rr}.6 & 1.0 \\ .3 & -.7\end{array}\right)$,
(c) $\left(\begin{array}{rrr}-1 & -1 & -4 \\ -2 & 0 & -4 \\ 1 & -1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 2 & 3 & -2 \\ -1 & -1 & 2\end{array}\right)$.

Key Concepts

Fixed Point

A fixed point in an iterative system is a state that remains unchanged when the system's transformation (or function) is applied. For a linear system defined by f(x) = Ax, the fixed point satisfies Ax = x. This concept is crucial to understand because fixed points often describe steady?state or equilibrium conditions in dynamical systems.

Eigenvalue and Eigenvector Correspondence

In linear algebra, fixed points of a linear transformation correspond to eigenvectors associated with the eigenvalue 1. This is because if Ax = x for a nonzero vector x, then x is an eigenvector of A with eigenvalue 1. Recognizing this relationship is key to analyzing the behavior of iterative processes.

Nullspace of (I-A) / Solving Homogeneous Equations

To find the fixed points of the system f(x) = Ax, one can rearrange the equation to (I - A)x = 0, where I is the identity matrix. The set of solutions (including possibly the trivial solution) is the nullspace of (I-A) and provides a systematic way of determining all fixed points of the linear system.

Linear Iterative Systems

Linear iterative systems involve repeated application of a coefficient matrix to a state vector. Their analysis, particularly determining fixed points and stability, rests on the properties of the matrix. Understanding how the coefficient matrix affects the evolution of the state vector is central to studying the convergence and dynamical behavior of these systems.

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