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A common misconception is that if A has a strictly
Chapter 5, Problem 21E(choose chapter or problem)
A common misconception is that if A has a strictly dominant eigenvalue, then, for any sufficiently large value of k, the vector \(A^{k} \mathbf{x}\) is approximately equal to an eigenvector of A. For the three matrices below, study what happens to \(A^{k} \mathbf{x}\) when \(\mathbf{x}=(.5, .5)\), and try to draw general conclusions (for a \(2 \times 2\) matrix).
a. \(A=\left[\begin{array}{ll}.8 & 0 \\ 0 & .2\end{array}\right]\)
b. \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & .8\end{array}\right]\)
c. \(A=\left[\begin{array}{ll}8 & 0 \\ 0 & 2\end{array}\right]\)
Questions & Answers
QUESTION:
A common misconception is that if A has a strictly dominant eigenvalue, then, for any sufficiently large value of k, the vector \(A^{k} \mathbf{x}\) is approximately equal to an eigenvector of A. For the three matrices below, study what happens to \(A^{k} \mathbf{x}\) when \(\mathbf{x}=(.5, .5)\), and try to draw general conclusions (for a \(2 \times 2\) matrix).
a. \(A=\left[\begin{array}{ll}.8 & 0 \\ 0 & .2\end{array}\right]\)
b. \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & .8\end{array}\right]\)
c. \(A=\left[\begin{array}{ll}8 & 0 \\ 0 & 2\end{array}\right]\)
ANSWER:Solution 21E
Step 1 of 7
The objective is to find the general conclusions for a matrix.
(a)
For compute
