Problem 21 - Advanced Engineering Mathematics

Question

(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

The eigenvalues are \(-2, -2, 2\). The orthogonal matrix constructed is \( \mathbf{P} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{pmatrix} \).

Step 1: Verify Eigenvectors

We start by checking if each column vector \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3 \) is an eigenvector of matrix \( \mathbf{A} \). An eigenvector of a matrix satisfies the equation \( \mathbf{A}\mathbf{K} = \lambda \mathbf{K} \), where \( \lambda \) is the eigenvalue.**Compute \( \mathbf{A}\mathbf{K}_1 \):**\[ \mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix} = \begin{pmatrix} 2 \times -1 + 2 \times 0 \ 2 \times 1 + 2 \times 0 \ 2 \times 1 + 2 \times (-1) \end{pmatrix} = \begin{pmatrix} -2 \ 2 \ 0 \end{pmatrix} \]This indicates \( \lambda_1 = -2 \).**Compute \( \mathbf{A}\mathbf{K}_2 \):**\[ \mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ 0 \ -1 \end{pmatrix} = \begin{pmatrix} 0 + 2 \times (-1) \ 2 \times 1 - 2 \times 1 \ 2 \times 1 \end{pmatrix} = \begin{pmatrix} -2 \ 2 \ -2 \end{pmatrix} \]This indicates \( \lambda_2 = -2 \).**Compute \( \mathbf{A}\mathbf{K}_3 \):**\[ \mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 0 & 2 & 2 \ 2 & 0 & 2 \ 2 & 2 & 0 \end{pmatrix} \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 4 \ 4 \ 4 \end{pmatrix} = 2 \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} \]This indicates \( \lambda_3 = 2 \).

Step 2: Identify Eigenvalues

Based on the previous calculations:- The eigenvector \( \mathbf{K}_1 \) corresponds to eigenvalue \( \lambda_1 = -2 \).- The eigenvector \( \mathbf{K}_2 \) corresponds to eigenvalue \( \lambda_2 = -2 \).- The eigenvector \( \mathbf{K}_3 \) corresponds to eigenvalue \( \lambda_3 = 4 \).

Step 3: Apply Gram-Schmidt

To construct an orthogonal matrix \( \mathbf{P} \), we apply the Gram-Schmidt process to the set of eigenvectors \( \{\mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3\} \).**Orthogonalize \( \mathbf{K}_1 \):**- Normalize to get \( \mathbf{u}_1 = \frac{\mathbf{K}_1}{\| \mathbf{K}_1 \|} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix} \)**Orthogonalize \( \mathbf{K}_2 \):**- Project \( \mathbf{K}_2 \) onto \( \mathbf{u}_1 \):\[ \text{Proj}_{\mathbf{u}_1}(\mathbf{K}_2) = \frac{\mathbf{K}_2 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1 = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \]- Subtract the projection from \( \mathbf{K}_2 \) and normalize:\[ \mathbf{u}_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \ 0 \ -1 \end{pmatrix} \]**Orthogonalize \( \mathbf{K}_3 \):**- Project \( \mathbf{K}_3 \) onto \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):\[ \mathbf{u}_3 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} \]The orthogonal matrix \( \mathbf{P} \) is:\[ \mathbf{P} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \-\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{pmatrix} \]

Step 4: Confirm Orthogonality and Normalize

Ensure that the vectors \( \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 \) are orthogonal and each has a unit norm. Compute dot products and norms:- \( \mathbf{u}_1 \cdot \mathbf{u}_2 = 0 \)- \( \mathbf{u}_1 \cdot \mathbf{u}_3 = 0 \)- \( \mathbf{u}_2 \cdot \mathbf{u}_3 = 0 \)Each vector is also unit norm:- \( \|\mathbf{u}_1\| = \|\mathbf{u}_2\| = \|\mathbf{u}_3\| = 1 \)Thus, \( \mathbf{P} \) is orthogonal.

Key Concepts

Eigenvalues

Eigenvalues are fundamental concepts in linear algebra. They are real or complex numbers that explain how a linear transformation changes the geometry of a space. Algebraically, if a matrix \( \mathbf{A} \) has a vector \( \mathbf{v} \) such that applying \( \mathbf{A} \) to \( \mathbf{v} \) stretches (or shrinks) it by a scalar (\( \lambda \)), then the vector \( \mathbf{v} \) is called an eigenvector, and the scalar \( \lambda \) is its eigenvalue.

When it comes to symmetric matrices, the eigenvalues have special characteristics: they are always real numbers. This is a significant property that can simplify many calculations and is a reason why symmetric matrices are highly valued in theory and practice.

To find the eigenvalues, we solve the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). In our exercise, after performing computations for the eigenvectors \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3 \), we identified the corresponding eigenvalues as \( -2, -2, \) and \( 2 \), respectively.

Symmetric Matrices

Symmetric matrices are a special type of matrix where the element \( a_{ij} \) is equal to the element \( a_{ji} \). This means that the matrix is equal to its transpose (\( \mathbf{A} = \mathbf{A}^T \)).

These matrices have several special properties:

• All eigenvalues are real, which simplifies analysis and computation.
• The eigenvectors of a symmetric matrix are orthogonal, meaning they are perpendicular to each other in their space.
• Symmetric matrices often appear in various applications such as physics, chemistry, and statistics, particularly because of their stability and simplicity.

For the given matrix \( \mathbf{A} \) in our problem, it's clear that \( \mathbf{A} \) is symmetric. This is because each element above the main diagonal (from top left to bottom right) aligns perfectly with the corresponding element below.

Gram-Schmidt Process

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, such as those spanned by the eigenvectors of a matrix.
This process takes a set of vectors, which might not be orthogonal, and tweaks them to form a set of orthogonal (at right angles to each other) vectors.

Here’s how you use the Gram-Schmidt process:

• Take the first vector as it is and normalize it (adjust its length to 1), creating your first orthogonal vector.
• For each subsequent vector, adjust it by removing components in the direction of all previously created orthogonal vectors.
• Normalize each adjusted vector to ensure it maintains a unit norm.

In our exercise, this process was applied to the eigenvectors \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3 \). The result is an orthogonal set of vectors, \( \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 \), which form the columns of the orthogonal matrix \( \mathbf{P} \).

Orthogonal Matrices

Orthogonal matrices are matrices whose columns (or rows) are orthonormal vectors. This means the vectors are both orthogonal to each other and have unit length.

Several key properties of orthogonal matrices make them very useful:

• Multiplying by an orthogonal matrix preserves vector norms and angles, making calculations more stable and reliable.
• The inverse of an orthogonal matrix is simply its transpose, which simplifies many computations.
• They also play a crucial role in rotating, reflecting, or projecting vectors in space without altering their lengths.

In our scenario, after applying the Gram-Schmidt process, we derived an orthogonal matrix \( \mathbf{P} \). Verifying orthogonality involves checking that the dot product of different columns is zero, while each column's norm is exactly one.