Problem 22 - 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. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{llll} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} -1 \\ 0 \\ 0 \\ 1 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 0 \\ 0 \end{array}\right), \quad \mathbf{K}_{4}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

Eigenvalues are 0 (for \(\mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3\)) and 4 (for \(\mathbf{K}_4\)). Construct matrix \(\mathbf{P}\) using Gram-Schmidt.

Step 1: Verify if Vectors are Eigenvectors

First, we need to verify that \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3, \mathbf{K}_4 \) are eigenvectors of the matrix \( \mathbf{A} \). This is done by checking if \( \mathbf{A} \mathbf{K} = \lambda \mathbf{K} \), where \( \lambda \) is the eigenvalue. Calculate \( \mathbf{A} \mathbf{K}_i \) for each vector \( \mathbf{K}_i \) and check if it results in a scalar multiple of \( \mathbf{K}_i \).

Step 2: Compute \(\mathbf{A}\mathbf{K}_1\)

Calculate \( \mathbf{A}\mathbf{K}_1 \): \[ \mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} -1 \ 0 \ 0 \ 1 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \ 0 \end{pmatrix} \] This shows that \( \mathbf{K}_1 \) corresponds to the eigenvalue \( \lambda = 0 \).

Step 3: Compute \(\mathbf{A}\mathbf{K}_2\)

Calculate \( \mathbf{A}\mathbf{K}_2 \): \[ \mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} -1 \ 0 \ 1 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \ 0 \end{pmatrix} \] This shows that \( \mathbf{K}_2 \) corresponds to the eigenvalue \( \lambda = 0 \).

Step 4: Compute \(\mathbf{A}\mathbf{K}_3\)

Calculate \( \mathbf{A}\mathbf{K}_3 \): \[ \mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} -1 \ 1 \ 0 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \ 0 \end{pmatrix} \] This shows that \( \mathbf{K}_3 \) corresponds to the eigenvalue \( \lambda = 0 \).

Step 5: Compute \(\mathbf{A}\mathbf{K}_4\)

Calculate \( \mathbf{A}\mathbf{K}_4 \): \[ \mathbf{A} \mathbf{K}_4 = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \ 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 4 \ 4 \ 4 \ 4 \end{pmatrix} \] This shows that \( \mathbf{K}_4 \) corresponds to the eigenvalue \( \lambda = 4 \).

Step 6: Orthogonalize Vectors using Gram-Schmidt

Now, use the Gram-Schmidt process to orthogonalize the set of eigenvectors \( \{ \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3, \mathbf{K}_4 \} \). The Gram-Schmidt process replaces a set of vectors with an orthogonal set having the same span. Begin with \( \mathbf{U}_1 = \mathbf{K}_1 \), then compute \( \mathbf{U}_i \) for \( i = 2, 3, 4 \) using orthogonal projection formulas.

Step 7: Normalize the Orthogonal Vectors

Normalize each orthogonal vector obtained from Gram-Schmidt to produce an orthonormal basis. An orthogonal matrix \( \mathbf{P} \) can be constructed where columns are these normalized vectors.

Step 8: Construct the Orthonormal Matrix \(\mathbf{P}\)

The columns of \( \mathbf{P} \) are the normalized vectors obtained from the previous step. Each vector should have a unit length, ensuring \( \mathbf{P} \) is an orthonormal matrix.

Key Concepts

Gram-Schmidt Process

The Gram-Schmidt process is a method used in linear algebra to orthogonalize a set of vectors in an inner product space. This means that the output from this process will be a set of vectors that are both linearly independent and perpendicular to each other. The main goal of Gram-Schmidt is to start with a set of vectors and transform them into a set of mutually orthogonal vectors that span the same space.

### How It Works
To apply the Gram-Schmidt process, follow these simple steps:

• Start with the first vector, which will be your initial orthogonal vector.
• For each subsequent vector, subtract the projection of the vector onto each of the previous orthogonal vectors using the projection formula.
• Repeat this process until all vectors are orthogonal to each other.

The projection formula for a vector is: \[\text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}\]This formula is used to find the component of one vector in the direction of another.

Remember, once the set is orthogonal, the vectors often need to be normalized (made into unit vectors) to create an orthonormal set. This is crucial for forming orthonormal matrices later.

Orthonormalization

Orthonormalization is the process where a set of vectors is not only made orthogonal but also each vector is normalized to have unit length. An orthonormal set has two essential properties: each vector is perpendicular to the others, and each vector has a magnitude of one. This is a critical concept because orthonormal vectors are simple to handle in calculations and transformations, especially for forming matrices.

### Why Orthonormalize?
Orthogonal vectors are already perpendicular, but normalization is key for various applications in linear algebra, such as:

• Ensuring stability in numerical computations.
• Facilitating easier projections and transformations because orthonormal bases preserve lengths and angles.

To normalize a vector, divide the vector by its magnitude (or length), which can be found using:\[||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}\]This ensures that the magnitude of the vector is 1, making it a unit vector.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose, meaning it remains unchanged when rows and columns are swapped. This property significantly simplifies analysis and computations, especially in the context of eigenvalues and eigenvectors.

### Properties of Symmetric Matrices
Symmetric matrices have several useful properties:

• Their eigenvalues are always real, which simplifies calculations and practical applications.
• The eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal.
• They can always be diagonalized by an orthogonal matrix, simplifying matrix equations considerably.

These properties make symmetric matrices particularly important in various fields such as physics and statistics, where they are used to represent covariance matrices and perform tasks like Principal Component Analysis (PCA).

Understanding symmetric matrices is essential when dealing with eigenvalues and eigenvectors, as their properties often lead to more straightforward solutions and intuitive insights into the problems at hand.