Problem 21 - Advanced Engineering Mathematics

Question

In Problems 21 and 22, (a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix and (b) identify the corresponding eigenvalues. (c) Proceed as in Example 4 and use the Gram-Schmidt process to construct an orthogonal matrix \(\mathbf{P}\) from the eigenvectors. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{lll} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right) \\ &\mathbf{K}_{2}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

Eigenvalues are \(-2\) for \( \mathbf{K}_1 \) and \( \mathbf{K}_2 \), and \(4\) for \( \mathbf{K}_3 \). Construct orthogonal matrix \( \mathbf{P} \) using Gram-Schmidt.

Step 1: Verify Eigenvector \( \mathbf{K}_1 \)

To verify that \( \mathbf{K}_1 \) is an eigenvector, calculate \( \mathbf{A} \mathbf{K}_1 \). \[ \mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix} = \begin{pmatrix} -2 \ 2 \ -2 \end{pmatrix} \] Check if it is a scalar multiple of \( \mathbf{K}_1 \): \( -2 \begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix} = \begin{pmatrix} -2 \ 2 \ 0 \end{pmatrix} \). Thus, \( \mathbf{K}_1 \) is an eigenvector with eigenvalue \(-2\).

Step 2: Verify Eigenvector \( \mathbf{K}_2 \)

Calculate \( \mathbf{A} \mathbf{K}_2 \).\[ \mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ 0 \ -1 \end{pmatrix} = \begin{pmatrix} -2 \ 2 \ 2 \end{pmatrix} \] Check if it is a scalar multiple of \( \mathbf{K}_2 \):\( -2 \begin{pmatrix} 1 \ 0 \ -1 \end{pmatrix} = \begin{pmatrix} -2 \ 0 \ 2 \end{pmatrix} \). Thus, \( \mathbf{K}_2 \) is an eigenvector with eigenvalue \(-2\).

Step 3: Verify Eigenvector \( \mathbf{K}_3 \)

Calculate \( \mathbf{A} \mathbf{K}_3 \).\[ \mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 4 \ 4 \ 4 \end{pmatrix} \] Check if it is a scalar multiple of \( \mathbf{K}_3 \):\( 4 \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 4 \ 4 \ 4 \end{pmatrix} \). Thus, \( \mathbf{K}_3 \) is an eigenvector with eigenvalue \(4\).

Step 4: Construct Orthogonal Matrix \( \mathbf{P} \) with Gram-Schmidt

Use the Gram-Schmidt process to orthogonalize \( \mathbf{K}_1 \), \( \mathbf{K}_2 \), and \( \mathbf{K}_3 \).**Step 1:** Normalize \( \mathbf{K}_1 \):\[ \mathbf{v}_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \-1 \0 \end{pmatrix} \]**Step 2:** Orthogonalize \( \mathbf{K}_2 \) with respect to \( \mathbf{v}_1 \):\[ \mathbf{v}_2 = \mathbf{K}_2 - \text{proj}_{\mathbf{v}_1}(\mathbf{K}_2) = \begin{pmatrix} 1 \0 \-1 \end{pmatrix} - \frac{1}{2} \begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix} = \begin{pmatrix} 0.5 \ 0.5 \ -1 \end{pmatrix} \]Normalize \( \mathbf{v}_2 \):\[ \mathbf{v}_2 = \frac{1}{\sqrt{1.5}} \begin{pmatrix} 0.5 \ 0.5 \ -1 \end{pmatrix} \]**Step 3:** Orthogonalize \( \mathbf{K}_3 \) with respect to \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \): Ensure orthogonality and normalize.**Step 4:** Normalize \( \mathbf{v}_3 \). The matrix \( \mathbf{P} \) is the collection of \( \mathbf{v}_1 \), \( \mathbf{v}_2 \), and \( \mathbf{v}_3 \) as columns.

Key Concepts

Symmetric Matrix

A symmetric matrix is a special type of square matrix that is identical to its transpose. This means that the matrix remains unchanged when you reflect it over its main diagonal. In mathematical terms, a matrix \( \mathbf{A} \) is symmetric if \( \mathbf{A} = \mathbf{A}^T \).
For example, take the matrix \( \mathbf{A} \) from the given exercise: \[ \begin{aligned} &\mathbf{A} = \left(\begin{array}{lll} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{array}\right) \end{aligned} \] This matrix is symmetric because swapping its rows and columns does not change the matrix.
Symmetric matrices have some interesting properties that make them special:

• Their eigenvalues are always real numbers.
• They can be diagonalized by an orthogonal matrix.
• They often show up in various branches of mathematics and engineering.

In practical terms, symmetric matrices are common when working with problems involving quadratic forms, physics, or statistics, where the symmetry often represents an underlying physical or mathematical property.

Gram-Schmidt Process

The Gram-Schmidt process is a mathematical method used to transform a set of vectors into an orthogonal set, which means that each vector in the set is perpendicular to the others.
This process is incredibly useful when you want to create an orthogonal basis from a set of vectors and is often used in linear algebra.
Here's how it works, using vectors \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3 \) from the exercise:

• Start by choosing one vector, say \( \mathbf{v}_1 = \mathbf{K}_1 \). You may need to normalize this vector to make it a unit vector, which involves dividing it by its length.
• Next, take the second vector \( \mathbf{K}_2 \) and project it onto the first vector \( \mathbf{v}_1 \). Subtract this projection from \( \mathbf{K}_2 \) to get a vector that is orthogonal to \( \mathbf{v}_1 \).
• This resulting vector can be normalized to give you \( \mathbf{v}_2 \). Repeat this process for any additional vectors, ensuring to orthogonalize against all previously processed vectors.

The main outcome of the Gram-Schmidt process is the creation of an orthogonal matrix \( \mathbf{P} \), where each column is a vector from the orthogonal set.
This process is fundamental in applications like QR decomposition, where a matrix is decomposed into a product of an orthogonal matrix and an upper triangular matrix, which is essential in solving linear systems.

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns and rows are both orthogonal unit vectors. This means that the transpose of the matrix is equal to its inverse: \( \mathbf{P}^T \mathbf{P} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix.
Orthogonal matrices have significant properties that make them quite powerful in linear transformations:

• They preserve the length of vectors, which means they do not change the vector's magnitude when applied.
• They are used to simplify complex mathematical problems, especially in eigenvalue problems, because orthogonal matrices can diagonalize symmetric matrices.
• This kind of matrix is crucial in coordinate transformations, such as rotating or reflecting coordinate systems.

In your exercise, after transforming the eigenvectors using the Gram-Schmidt process, you constructed an orthogonal matrix \( \mathbf{P} \). Each column of \( \mathbf{P} \) was an eigenvector from the original set, now orthogonalized.
Orthogonal matrices are extremely practical and often make computations not just possible but also more efficient, particularly in numerical algorithms.