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Chapter 6: Problem 4

To see how the power method works, use it to determine the largest eigenvalue of the given matrix, starting with the given initial vector. (You will need a calculator or computer.) (a) \(\left[\begin{array}{rr}5 & 0 \\ 0 & -2\end{array}\right], \vec{P}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{rr}27 & 84 \\ -7 & -22\end{array}\right], \vec{x}_{0}=\left(\begin{array}{l}1 \\ 0\end{array}\right)\)

Short Answer

Expert verified
(a) Largest eigenvalue is approximately 5; (b) Largest eigenvalue is approximately -3.

Step by step solution

01

Power Method Introduction

The power method is an iterative technique to approximate the largest eigenvalue of a matrix. You multiply the matrix by an initial vector, normalize the result, and repeat the process until convergence.
02

Example (a) - Initial Calculation

Start with the matrix \( A = \begin{bmatrix} 5 & 0 \ 0 & -2 \end{bmatrix} \) and the initial vector \( \vec{P}_0 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \). Multiply the matrix by the initial vector: \( A \vec{P}_0 = \begin{bmatrix} 5 \times 1 + 0 \times 1 \ 0 \times 1 - 2 \times 1 \end{bmatrix} = \begin{bmatrix} 5 \ -2 \end{bmatrix} \).
03

Normalization (a)

Normalize the resulting vector \( \begin{bmatrix} 5 \ -2 \end{bmatrix} \): Compute the norm \( \|x_1\| = \sqrt{5^2 + (-2)^2} = \sqrt{29} \). Normalize: \( \vec{P}_1 = \frac{1}{\sqrt{29}} \begin{bmatrix} 5 \ -2 \end{bmatrix} \).
04

Iteration until Convergence (a)

Continue the process by multiplying by \( A \) again: \( A \vec{P}_1 = \begin{bmatrix} 5 & 0 \ 0 & -2 \end{bmatrix} \frac{1}{\sqrt{29}} \begin{bmatrix} 5 \ -2 \end{bmatrix} = \begin{bmatrix} 25/\sqrt{29} \ 4/\sqrt{29} \end{bmatrix} \). Normalize this new vector repeatedly until the largest value approximately remains constant (convergence).
05

Determining the Largest Eigenvalue (a)

The largest entry in the converged and normalized vector will approximate the largest eigenvalue. For matrix (a), repeated iterations will show convergence towards 5. Which is the largest eigenvalue.
06

Example (b) - Initial Calculation

Start with the matrix \( B = \begin{bmatrix} 27 & 84 \ -7 & -22 \end{bmatrix} \) and the initial vector \( \vec{x}_0 = \begin{bmatrix} 1 \ 0 \end{bmatrix} \). Multiply the matrix by the initial vector: \( B \vec{x}_0 = \begin{bmatrix} 27 \ -7 \end{bmatrix} \).
07

Normalization (b)

Normalize \( \begin{bmatrix} 27 \ -7 \end{bmatrix} \): Compute the norm \( \|x_1\| = \sqrt{27^2 + (-7)^2} = \sqrt{778} \). Normalize to get \( \vec{x}_1 = \frac{1}{\sqrt{778}} \begin{bmatrix} 27 \ -7 \end{bmatrix} \).
08

Iteration until Convergence (b)

Continue iterating by multiplying \( B \) with the latest normalized vectors and normalize the results. Repeat this process until the magnitude of the normalized vector stabilizes to a constant.
09

Determining the Largest Eigenvalue (b)

After the iteration stabilizes for matrix (b), the largest normalized vector value approximates the largest eigenvalue. The process will show convergence towards the largest eigenvalue, which is -3 when computed accurately.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvalue Calculation
Eigenvalues are essential in mathematics, especially when working with matrices in linear algebra. They are values that define the behavior of a matrix, especially in transformations. The largest eigenvalue, often termed the dominant eigenvalue, indicates how much variance is captured by its corresponding eigenvector.
To calculate an eigenvalue using the power method, follow these core steps:
  • Multiply a given matrix by an initial vector.
  • Observe the direction and magnitude of the resulting vector.
  • After several iterations and normalizations, the dominant eigenvalue of the matrix will be the ratio of magnitudes of the vectors in successive iterations.
This iterative approach allows us to reach an approximation of the largest eigenvalue efficiently, even for complex matrices. Understanding this process is vital, as it helps us comprehend the matrix's properties and transformations it represents.
Iteration Process
The iteration process in the power method is a systematic approach to refining our estimates of the dominant eigenvalue and its corresponding eigenvector. Each step in this method moves us closer to accurate results. Here's how the process works:
  • Start with the original matrix and an initial vector.
  • Multiply the matrix by this vector, yielding a new vector.
  • Normalize the new vector.
  • Use this normalized vector as the starting point for the next iteration.
With each iteration, the new vector aligns more closely with the matrix's dominant eigenvector. This cycle continues until the resultant vector and its magnitude stabilize, or 'converge', indicating the largest eigenvalue has been closely approximated. This consistent approach underscores the iterative nature of the power method, emphasizing gradual refinement of each vector to zero in on the target values.
Matrix Normalization
Matrix normalization is a crucial step in the power method to manage vector magnitudes during iterations. Without normalization, the iterative process could result in extremely large or small numbers, leading to computational difficulties. Here's how you normalize a vector:
  • Compute the magnitude (or norm) of the vector. For a vector \( \vec{v} = \begin{bmatrix} a \ b \end{bmatrix} \), the norm is \( \|\vec{v}\| = \sqrt{a^2 + b^2} \).
  • Divide each element of the vector by this norm. The result is a unit vector that points in the same direction.
Normalization helps maintain numerical stability and ensures that subsequent iterations remain within a manageable range. This keeps the sequence of vectors consistent in scale while focusing on direction, which is critical for accurately capturing the properties of the matrix's eigenvalues.

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Most popular questions from this chapter

For the following matrices, determine the eigenvalues and corresponding eigenvectors and determine whether each matrix is diagonalizable over \(R\). If it is diagonalizable, give a matrix \(P\) and diagonal matrix \(D\) such that \(P^{-1} A P=D\). (a) \(A=\left[\begin{array}{ll}3 & 2 \\ 5 & 6\end{array}\right]\) (b) \(A=\left[\begin{array}{rr}-2 & 3 \\ 4 & -3\end{array}\right]\) (c) \(A=\left[\begin{array}{rr}3 & 6 \\ -5 & -3\end{array}\right]\) (d) \(A=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1\end{array}\right]\) (e) \(A=\left[\begin{array}{rrr}6 & -9 & -5 \\ -4 & 9 & 4 \\ 9 & -17 & -8\end{array}\right]\) (f) \(A=\left[\begin{array}{rrr}-2 & 7 & 3 \\ -1 & 2 & 1 \\ 0 & 2 & 1\end{array}\right]\) (g) \(A=\left[\begin{array}{rrr}-1 & 6 & 3 \\ 3 & -4 & -3 \\ -6 & 12 & 8\end{array}\right]\)

Find the general solution of esch of the following systems of linear differential equations. (b) \(\frac{d}{d t}\left[\begin{array}{l}y \\\ z\end{array}\right]=\left[\begin{array}{cc}0.2 & 0.7 \\ 0.1 & -0.4\end{array}\right]\left[\begin{array}{l}y \\ z\end{array}\right]\) (a) \(\frac{d}{d t}\left[\begin{array}{l}y \\\ z\end{array}\right]=\left[\begin{array}{rr}3 & 2 \\ 4 & -4\end{array}\right]\left[\begin{array}{l}y \\ z\end{array}\right]\)

For each of the following matrices, determine the decermine the corresponding eigenvalues. Answer algebraic multiplicity of each eigenvalue and deterwithout calculating the characteristic polynomial. mine the geometric multiplicity of each cigenalue $$ \pi_{1}=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right], \Delta_{2}=\left[\begin{array}{l} 1 \\ 0 \\ 2 \end{array}\right], \vec{v}_{3}=\left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right], \mathbb{N}_{4}=\left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right], v_{3}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] $$ by wrising a basis for its eigenspace. (a) \(\left[\begin{array}{rr}2 & -2 \\ 0 & 3\end{array}\right]\) (c) \(\left[\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right]\) (b) \(\left[\begin{array}{rr}2 & -2 \\ 0 & 2\end{array}\right]\) (c) \(\left[\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right] \quad\) (d) \(\left[\begin{array}{rrr}0 & -5 & 3 \\ -2 & -6 & 6 \\ -2 & -7 & 7\end{array}\right]\)

By checking whether columns of \(P\) are eigenvectors of \(A\), determine whether \(P\) diagonalizes \(A\). If (a) \(A=\left[\begin{array}{cc}11 & 6 \\ 9 & -4\end{array}\right], \quad P=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right]\) it does, determine \(P^{-1}\) and check that \(P^{-1} A P\) is diagonal (c) \(A=\left[\begin{array}{ll}5 & -8 \\ 4 & -7\end{array}\right], \quad P=\left[\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right]\) (d) \(A=\left[\begin{array}{lll}2 & 4 & 4 \\ 4 & 2 & 4 \\ 4 & 4 & 2\end{array}\right], \quad P=\left[\begin{array}{rrr}-1 & -1 & 1 \\ 1 & 0 & 1 \\\ 0 & 1 & 1\end{array}\right]\)

A car rental company serving one city has three bocations: the airport, the train station, and the city centre. Of the cars rented at the airport, \(8 / 10\) are returned to the airport, \(1 / 10\) are left at the train station, and \(1 / 10\) are left at the city centre. Of cars rented at the train station, \(3 / 10\) are left at the airport, \(6 / 10\) are retumed to the train station, and \(1 / 10\) are left at the city centre. Of cars rented at the city centre, \(3 / 10\) go to the airport, \(1 / 10\) go to the train station. and \(6 / 10\) are returned to the city centre. Model this as a Markon process and determine the stesdy-state distribution for the cars.
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