Question

In each case, find the exact eigenvalues and determine corresponding eigenvectors. Then start with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 1\end{array}\right]\) and compute \(\mathbf{x}_{4}\) and \(r_{3}\) using the power method. a. \(A=\left[\begin{array}{rr}2 & -4 \\ -3 & 3\end{array}\right]\) b. \(A=\left[\begin{array}{rr}5 & 2 \\ -3 & -2\end{array}\right]\) c. \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\) d. \(A=\left[\begin{array}{ll}3 & 1 \\ 1 & 0\end{array}\right]\)

Short answer

Solve the characteristic equation to find eigenvalues, eigenvectors. Use initial vector in power method to compute \(\mathbf{x}_4\) and \(r_3\).

Step-by-step solution

Compute the Eigenvalues of Matrix A

For matrix \(A\), the eigenvalues can be found by solving the characteristic equation \(\det(A - \lambda I) = 0\). For example, for matrix \(A = \begin{bmatrix} 2 & -4 \ -3 & 3 \end{bmatrix}\), compute the determinant as follows: \(\det\begin{pmatrix} 2-\lambda & -4 \ -3 & 3-\lambda \end{pmatrix} = (2-\lambda)(3-\lambda) - (-4)(-3) = \lambda^2 - 5\lambda - 6\). Solve \(\lambda^2 - 5\lambda - 6 = 0\) using factoring or the quadratic formula to find \(\lambda_1\) and \(\lambda_2\).

Find Corresponding Eigenvectors

Once you have the eigenvalues, find the eigenvectors by solving \((A - \lambda I)\mathbf{v} = 0\) for each eigenvalue \(\lambda\). Continue with matrix \(A = \begin{bmatrix} 2 & -4 \ -3 & 3 \end{bmatrix}\) and eigenvalue \(\lambda_1\), set up the equation \((A - \lambda_1 I)\mathbf{v} = 0\). Solve this system to find \(\mathbf{v}_1\), a non-zero vector (eigenvector) corresponding to \(\lambda_1\). Repeat for \(\lambda_2\) to find \(\mathbf{v}_2\).

Apply the Power Method to Compute \(\mathbf{x}_4\)

Initialize with \(\mathbf{x}_0 = \begin{bmatrix} 1 \ 1 \end{bmatrix}\). Compute subsequent vectors using \(\mathbf{x}_{k+1} = A\mathbf{x}_k\). At each step, normalize the results to prevent overflow using \(\mathbf{x}_{k+1} = \frac{A\mathbf{x}_k}{\|A\mathbf{x}_k\|}\). Continue the iteration for four steps (\(\mathbf{x}_1\) to \(\mathbf{x}_4\)).

Compute the Rayleigh Quotient \(r_3\)

The Rayleigh quotient after three iterations is computed as a form of the eigenvalue approximation and is given by \(r_k = \frac{\mathbf{x}_k^T A \mathbf{x}_k}{\mathbf{x}_k^T \mathbf{x}_k}\). For \(r_3\), use \(\mathbf{x}_3\) and the matrix \(A\) to calculate \(r_3\). Substitute \(\mathbf{x}_3\) into the formula to get the value.

Key concepts

Characteristic Equation

The characteristic equation is a foundational concept in finding eigenvalues of a matrix. It involves a polynomial equation that arises from the determinant of the matrix equation \( A - \lambda I \). To find the eigenvalues of a matrix, you solve this determinant set to zero: \( \det(A - \lambda I) = 0 \).
Let's break this down a bit:

• First, \( A \) is your matrix, \( \lambda \) is a scalar (representing the eigenvalues), and \( I \) is the identity matrix of the same size as \( A \).
• The subtraction \( A - \lambda I \) modifies the diagonal elements of \( A \) by subtracting \( \lambda \).
• Calculate the determinant of this new matrix \( A - \lambda I \), which transforms the problem into a polynomial.

For example, with matrix \( A = \begin{bmatrix} 2 & -4 \ -3 & 3 \end{bmatrix} \), its characteristic equation is derived as \( \lambda^2 - 5\lambda - 6 = 0 \), after performing these steps.
Solving this equation gives us the eigenvalues, which are crucial for further steps like finding eigenvectors or using numerical methods.

Power Method

The power method is an iterative technique to approximate the largest eigenvalue and its associated eigenvector for a matrix. This is especially useful for large matrices where direct computation of eigenvalues is not feasible.
Here's how the power method works:

• Start with an initial guess for the eigenvector, commonly \( \mathbf{x}_0 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), as given in the problem.
• Multiply the current vector by the matrix \( A \) to get a new vector: \( \mathbf{x}_{k+1} = A\mathbf{x}_k \).
• Normalize \( \mathbf{x}_{k+1} \) to prevent numerical instability, typically by dividing by its norm \( \|A\mathbf{x}_k\| \).
• Repeat this process several times to refine the approximation.

Through iteration, the vector \( \mathbf{x}_k \) will approximate the eigenvector corresponding to the dominant eigenvalue. Since we're focusing on \( \mathbf{x}_4 \), applying these steps four times refines the approximation, providing an estimate for the largest eigenvalue.

Rayleigh Quotient

The Rayleigh Quotient is a formula used to approximate an eigenvalue for a given matrix and vector. It provides an efficient way to gauge the accuracy of the power method's approximation.
The formula for the Rayleigh quotient is:

• \( r_k = \frac{\mathbf{x}_k^T A \mathbf{x}_k}{\mathbf{x}_k^T \mathbf{x}_k} \)

Here, \( \mathbf{x}_k \) is the current approximation of the eigenvector.
This quotient is particularly useful because:

• It converges to the eigenvalue of the matrix corresponding to the eigenvector as \( \mathbf{x}_k \) converges.
• Using the result of the power method at iteration \( \mathbf{x}_3 \), you can quickly approximate the corresponding eigenvalue with \( r_3 \).

In the example problem, after calculating \( \mathbf{x}_3 \) through the power method, inserting it into the Rayleigh quotient formula gives \( r_3 \), providing insight into how close \( \lambda_1 \) or \( \lambda_2 \) has been approximated.