Problem 2 - 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. $$ \left(\begin{array}{rrr} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array}\right) ;\left(\begin{array}{r} -2 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) $$

Step-by-Step Solution

Answer

Eigenvectors: \((-2, 1), (0, 2), (1, -1)\) with orthogonal vectors.

Step 1: Eigenvectors Verification for First Vector

For the matrix \[A = \begin{pmatrix} 1 & -1 & -1 \-1 & 1 & -1 \-1 & -1 & 1\end{pmatrix}\]check if \(\mathbf{v}_1 = \begin{pmatrix} -2 \ 1 \ 1 \end{pmatrix}\) is an eigenvector. Compute \[ A\mathbf{v}_1 = \begin{pmatrix} 1 & -1 & -1 \ -1 & 1 & -1 \ -1 & -1 & 1 \end{pmatrix}\begin{pmatrix} -2 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} -4 \ 2 \ 2 \end{pmatrix} = -2 \mathbf{v}_1\]. Since \(A\mathbf{v}_1 = -2 \mathbf{v}_1\), \(\mathbf{v}_1\) is an eigenvector with eigenvalue \(-2\).

Step 2: Eigenvectors Verification for Second Vector

Check if \(\mathbf{v}_2 = \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix}\) is an eigenvector. Compute \[ A\mathbf{v}_2 = \begin{pmatrix} 1 & -1 & -1 \ -1 & 1 & -1 \ -1 & -1 & 1 \end{pmatrix}\begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} = \begin{pmatrix} 0 \ 2 \ -2 \end{pmatrix} = 2\begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} = 2\mathbf{v}_2 \]. Thus, \(\mathbf{v}_2\) is an eigenvector with eigenvalue \(2\).

Step 3: Eigenvectors Verification for Third Vector

Check if \(\mathbf{v}_3 = \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix}\) is an eigenvector. Compute \[ A\mathbf{v}_3 = \begin{pmatrix} 1 & -1 & -1 \ -1 & 1 & -1 \ -1 & -1 & 1 \end{pmatrix}\begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} -1 \ -1 \ -1 \end{pmatrix} = -1\begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = -1\mathbf{v}_3 \]. Therefore, \(\mathbf{v}_3\) is an eigenvector with eigenvalue \(-1\).

Step 4: Orthogonality Check for Eigenvectors

To check that the eigenvectors are orthogonal, compute their dot products. For \(\mathbf{v}_1\) and \(\mathbf{v}_2\):\[\mathbf{v}_1 \cdot \mathbf{v}_2 = \begin{pmatrix} -2 \ 1 \ 1 \end{pmatrix} \cdot \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} = (-2)(0) + (1)(1) + (1)(-1) = 0\]. They are orthogonal.For \(\mathbf{v}_1\) and \(\mathbf{v}_3\):\[\mathbf{v}_1 \cdot \mathbf{v}_3 = \begin{pmatrix} -2 \ 1 \ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = (-2)(1) + (1)(1) + (1)(1) = 0\]. They are orthogonal.For \(\mathbf{v}_2\) and \(\mathbf{v}_3\):\[\mathbf{v}_2 \cdot \mathbf{v}_3 = \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = (0)(1) + (1)(1) + (-1)(1) = 0\]. They are orthogonal. All vector pairs are orthogonal.

Key Concepts

Symmetric Matrices

A symmetric matrix is a special type of square matrix that is symmetrical about its main diagonal, meaning the matrix is equal to its transpose. For any symmetric matrix \( A \), this means that \( A = A^T \). Symmetric matrices have several important mathematical properties that make them valuable, especially in the context of eigenvectors and eigenvalues.
One of the key characteristics of symmetric matrices is that they have real eigenvalues. This is incredibly advantageous because it simplifies calculations and helps ensure the stability of numerical algorithms. Moreover, the eigenvectors associated with distinct eigenvalues of a symmetric matrix are orthogonal.
Symmetric matrices often appear in physics and engineering problems—for instance, when dealing with stress and strain in materials. They are crucial in transforming bases in vector spaces and solving optimization problems as well.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors and operations on vectors, including those involving matrices. It provides the tools to describe systems of linear equations, and it's foundational in understanding various mathematical phenomena and practical applications.
Matrix operations, like multiplication, are fundamental in linear algebra. They allow us to transform vectors and compute various properties of data structures. Understanding how to compute eigenvectors and eigenvalues contributes to numerous applications, such as in computer graphics and machine learning.
Linear algebra is essential in many fields, including computer science, statistics, and physics. It enables professionals to model and solve real-world problems by transforming complex systems into manageable mathematical equations.

• Manipulating matrices
• Understanding vector spaces
• Solving systems of equations

are just a snippet of what linear algebra offers.

Orthogonality of Vectors

Orthogonality in vectors refers to a situation where two or more vectors are perpendicular to each other. In mathematical terms, this means their dot product is zero. If vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) are orthogonal, then \( \mathbf{v}_1 \cdot \mathbf{v}_2 = 0 \).
The concept of orthogonality is fundamental in the realm of linear algebra and has numerous applications. It's crucial when dealing with eigenvectors of symmetric matrices, as these are often orthogonal when their eigenvalues are distinct, simplifying the diagonalization process.
Additionally, orthogonal vectors are extremely useful in fields like signal processing, where orthogonal functions are used to separate signals into different frequency components. In statistics, orthogonal vectors help in techniques such as Principal Component Analysis (PCA), which reduces the dimensionality of data by removing correlations.