Problem 1 - Advanced Engineering Mathematics

Question

In Problems \(1-4\), (a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix, (b) identify the corresponding eigenvalues, and (c) verify that the column vectors are orthogonal. $$ \left(\begin{array}{rrr} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 15 \end{array}\right) ; \quad\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 4 \\ 0 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ 0 \\ -4 \end{array}\right) $$

Step-by-Step Solution

Answer

Eigenvectors are verified with eigenvalues \(-4\), \(1\), and \(16\). Vectors are orthogonal.

Step 1: Verify Eigenvector for Vector 1

Given vector \( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \), multiply by the matrix \( A = \begin{pmatrix} 0 & 0 & -4 \ 0 & -4 & 0 \ -4 & 0 & 15 \end{pmatrix} \). Calculating: \\[ A \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ -4 \ 0 \end{pmatrix} = -4 \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}. \] \Thus, \( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \) is an eigenvector with eigenvalue \(-4\).

Step 2: Verify Eigenvector for Vector 2

Given vector \( \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} \), multiply by the matrix \( A \). Calculating: \\[ A \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} = \begin{pmatrix} -4 \ 0 \ 31 \end{pmatrix}. \] \This is equivalent to \( 1 \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} \), hence \( \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} \) is an eigenvector with eigenvalue \(1\).

Step 3: Verify Eigenvector for Vector 3

Given vector \( \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} \), multiply by the matrix \( A \). Calculating: \\[ A \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} = \begin{pmatrix} 16 \ 0 \ -61 \end{pmatrix}. \] \This is equivalent to \( 16 \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} \), indicating an eigenvalue of \(16\).

Step 4: Verify Orthogonality

To verify orthogonality, confirm that the dot products of distinct pairs of the vectors are zero. \1. Compute \( \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} = 0\). \2. Compute \( \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} = 0\). \3. Compute \( \begin{pmatrix} 4 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} = 0\). \All dot products are zero, verifying that the vectors are orthogonal.

Key Concepts

Eigenvalues

Understanding eigenvalues is vital when solving problems involving matrices, especially in linear algebra. An eigenvalue is a special number associated with a given vector and matrix. When you multiply a matrix by one of its eigenvectors, the result is the eigenvector itself scaled by the eigenvalue. This can be mathematically represented as:\[\mathbf{A} \mathbf{v} = \lambda \mathbf{v}\]where:

• \( \mathbf{A} \) is the matrix in question,
• \( \mathbf{v} \) is the eigenvector,
• \( \lambda \) is the eigenvalue.

In the exercise, for the first vector \( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \), the matrix scales it by \(-4\), showing that \(-4\) is an eigenvalue. A similar calculation for the other vectors reveals their corresponding eigenvalues \(1\) and \(16\). Eigenvalues tell us how vectors are stretched or shrunk when the matrix is applied, without changing their direction.

Symmetric Matrix

A symmetric matrix is a square matrix that is identical to its transpose, meaning \( A = A^T \). This characteristic is crucial because symmetric matrices have properties that simplify many problems in linear algebra. One notable feature of symmetric matrices is that their eigenvalues are always real numbers.The symmetric matrix in the exercise:\[\begin{pmatrix}0 & 0 & -4 \0 & -4 & 0 \-4 & 0 & 15\end{pmatrix}\]possesses this property, where swapping rows and columns doesn't change the matrix.Symmetric matrices are important because:

• They have orthogonal eigenvectors, ensuring easy computations in various applications.
• They appear in diverse fields such as physics, statistics, and engineering due to their stability and predictability.
Understanding symmetric matrices helps in grasping the broader aspects of matrix behavior and aids in simplifying complex problems.

Orthogonality

Orthogonality in vectors means that they are perpendicular to each other. In more mathematical terms, two vectors are orthogonal if their dot product equals zero. This is a crucial concept in linear algebra, as orthogonal vectors simplify many calculations.In the exercise, to verify orthogonality, you take the dot product of every pair of vectors provided. For example:

• \( \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 4 \ 0 \ 1 \end{pmatrix} = 0 \)
• \( \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} = 0 \)
• \( \begin{pmatrix} 4 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 0 \ -4 \end{pmatrix} = 0 \)

All dot products turn out to be zero, verifying their orthogonality. Having orthogonal vectors facilitates vector decomposition and transformation, crucial in dimensionality reduction techniques like Principal Component Analysis.