Problem 14 - Advanced Engineering Mathematics

Question

In Problems 11-18, proceed as in Example 3 to construct an orthogonal matrix from the eigenvectors of the given symmetric matrix. (The answers are not unique.) $$ \left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

Construct eigenvectors, normalize them, and form the matrix with these vectors.

Step 1: Find Eigenvalues

Given the matrix \( A = \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix} \), we first find its characteristic equation. The matrix can be written as \( A - \lambda I = \begin{pmatrix} 1-\lambda & 1 \ 1 & -\lambda \end{pmatrix} \). Compute the determinant: \( (1-\lambda)(-\lambda) - 1 = -\lambda^2 + \lambda - 1 =0 \). Solving this quadratic equation, we find the eigenvalues \( \lambda_1 = \frac{1+\sqrt{5}}{2} \) and \( \lambda_2 = \frac{1-\sqrt{5}}{2} \).

Step 2: Find Eigenvectors

Substitute \( \lambda_1 = \frac{1+\sqrt{5}}{2} \) back into \( A - \lambda I \), resulting in \( \begin{pmatrix} (1-\lambda_1) & 1 \ 1 & -\lambda_1 \end{pmatrix} \). Solving \( \begin{pmatrix} -\frac{\sqrt{5}}{2} & 1 \ 1 & -\frac{1+\sqrt{5}}{2} \end{pmatrix}\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix} \), we get the eigenvector \( v_1 = \begin{pmatrix} 1 \ \frac{1-\sqrt{5}}{2} \end{pmatrix} \). Similarly, substitute \( \lambda_2 = \frac{1-\sqrt{5}}{2} \) and find \( v_2 = \begin{pmatrix} 1 \ \frac{1+\sqrt{5}}{2} \end{pmatrix} \).

Step 3: Normalize Eigenvectors

Normalize the eigenvectors found in Step 2. For \( v_1 \,=\, \begin{pmatrix} 1 \ \frac{1-\sqrt{5}}{2} \end{pmatrix} \), the norm is \( \sqrt{1^2 + \left(\frac{1-\sqrt{5}}{2}\right)^2} \). After calculating, \( v_{1\,norm} = \frac{v_1}{||v_1||} \). Do the same for \( v_2 \), resulting in \( v_{2\,norm} = \frac{v_2}{||v_2||} \).

Step 4: Construct Orthogonal Matrix

With normalized eigenvectors \( v_{1\,norm} \) and \( v_{2\,norm} \), construct the orthogonal matrix \( Q = \begin{pmatrix} v_{1\,norm} & v_{2\,norm} \end{pmatrix} \). Since eigenvectors from symmetric matrices are orthogonal, \( Q \) will be an orthogonal matrix.

Key Concepts

Eigenvalues

The eigenvalues of a matrix provide deep insights into its characteristics, especially when it is symmetric. To find the eigenvalues, you construct the characteristic equation of the matrix. This involves calculating the determinant of the matrix subtracted by the eigenvalue times the identity matrix, denoted as \( A - \lambda I \). For the matrix in the problem, the determinant equals \( -\lambda^2 + \lambda - 1 \). Solving such a quadratic equation gives you the eigenvalues \( \lambda_1 = \frac{1+\sqrt{5}}{2} \) and \( \lambda_2 = \frac{1-\sqrt{5}}{2} \).
Eigenvalues help in simplifying the matrix into diagonal form, which makes many operations easier. They reveal key information about the matrix, such as stability and oscillatory behavior. For instance, these values show that the transformations represented by this matrix scale and rotate vectors in a specific manner.

Eigenvectors

Eigenvectors are vectors that, when acted upon by a matrix, change only in scale, not direction. For symmetric matrices, eigenvectors corresponding to different eigenvalues are perpendicular to each other. To find an eigenvector, substitute an eigenvalue back into \( A - \lambda I \) and solve the resulting system of equations.

• For \( \lambda_1 = \frac{1+\sqrt{5}}{2} \), the solution yields \( v_1 = \begin{pmatrix} 1 & \frac{1-\sqrt{5}}{2} \end{pmatrix} \).
• Similarly, for \( \lambda_2 = \frac{1-\sqrt{5}}{2} \), we find \( v_2 = \begin{pmatrix} 1 & \frac{1+\sqrt{5}}{2} \end{pmatrix} \).

Since the matrix is symmetric, its eigenvectors are orthogonal, providing a foundation for constructing an orthogonal matrix. These vectors are crucial because they provide the axes along which the transformation described by the matrix acts straightforwardly - purely scaling along these directions.

Symmetric Matrix

A symmetric matrix is one where the transpose of the matrix equals the original matrix, meaning \( A^T = A \). This characteristic ensures several mathematical properties which simplify analysis. The given matrix \( \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix} \) is symmetric because the transpose doesn't alter its structure. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. These features make them important in various fields like physics, where they relate to energy and stability systems. They also allow for diagonalization, making computations less complicated as operations become scalar, not matrix-based. Finding orthogonal eigenvectors directly stems from this symmetry property, ensuring that these vectors span the entire space effectively and neatly.

Normalization

Normalization refers to the process of adjusting the values of eigenvectors so they have a unit length or norm of 1. This step is crucial in ensuring that the matrix composed of eigenvectors remains orthogonal. To normalize a vector \( v \), divide each component by its norm, denoted as \( ||v|| \).

• For the eigenvector \( v_1 = \begin{pmatrix} 1 & \frac{1-\sqrt{5}}{2} \end{pmatrix} \), calculate its norm and adjust the vector accordingly.
• Repeat the process for \( v_2 \).

This normalization does not change the direction of eigenvectors, only their magnitude, allowing the orthogonal matrix construction by aligning length and ensuring orthogonality. The matrix composed of these normalized vectors, called an orthogonal matrix, has desirable properties like preserving distances and angles, making it unitary in mathematical terms.