Problem 4 - Advanced Engineering Mathematics

Question

In Problems, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A} .\) Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} 2 & 8 \\ -1 & -2 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{l} 0 \\ 0 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{c} 2+2 i \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{c} 2+2 i \\ 1 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

None of \( \mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3 \) are eigenvectors of \( \mathbf{A} \).

Step 1: Define Eigenvector and Eigenvalue

An eigenvector \( \mathbf{v} \) of a matrix \( \mathbf{A} \) is a non-zero vector such that \( \mathbf{Av} = \lambda \mathbf{v} \), where \( \lambda \) is a scalar called the eigenvalue. We need to check if each given vector satisfies this equation with respect to \( \mathbf{A} \).

Step 2: Evaluate \( \mathbf{K}_1 \) as an Eigenvector

Check if \( \mathbf{K}_1 = \begin{pmatrix} 0 \ 0 \end{pmatrix} \) is an eigenvector. Calculate \( \mathbf{A} \mathbf{K}_1 \): \[\mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 2 & 8 \ -1 & -2 \end{pmatrix} \begin{pmatrix} 0 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}.\]Since eigenvectors cannot be zero vectors by definition, \( \mathbf{K}_1 \) is not an eigenvector.

Step 3: Evaluate \( \mathbf{K}_2 \) as an Eigenvector

Check if \( \mathbf{K}_2 = \begin{pmatrix} 2+2i \ -1 \end{pmatrix} \) is an eigenvector. Calculate \( \mathbf{A} \mathbf{K}_2 \):\[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 2 & 8 \ -1 & -2 \end{pmatrix} \begin{pmatrix} 2+2i \ -1 \end{pmatrix} = \begin{pmatrix} (2)(2+2i) + (8)(-1) \ (-1)(2+2i) + (-2)(-1) \end{pmatrix}.\]Simplifying this gives:\[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 4 + 4i - 8 \ -2 - 2i + 2 \end{pmatrix} = \begin{pmatrix} -4 + 4i \ -2i \end{pmatrix}.\]We need to see if this result is a scalar multiple of \( \mathbf{K}_2 \). Let's compare by writing \(-1(2+2i) = -2 - 2i\) and \(-1(-1) = 1\), so \(-1 \mathbf{K}_2= \begin{pmatrix} -4 + 4i \ 1 \end{pmatrix}\) which does not match. Hence, \( \mathbf{K}_2 \) is not an eigenvector.

Step 4: Evaluate \( \mathbf{K}_3 \) as an Eigenvector

Check if \( \mathbf{K}_3 = \begin{pmatrix} 2+2i \ 1 \end{pmatrix} \) is an eigenvector. Calculate \( \mathbf{A} \mathbf{K}_3 \):\[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 2 & 8 \ -1 & -2 \end{pmatrix} \begin{pmatrix} 2+2i \ 1 \end{pmatrix} = \begin{pmatrix} (2)(2+2i) + 8(1) \ (-1)(2+2i) + (-2)(1) \end{pmatrix}.\]Simplifying this gives:\[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 4 + 4i + 8 \ -2 - 2i - 2 \end{pmatrix} = \begin{pmatrix} 12 + 4i \ -4 - 2i \end{pmatrix}.\]Now find \( \lambda \) such that \( \mathbf{A} \mathbf{K}_3 = 3(2+2i) \mathbf{K}_3 = 3 \begin{pmatrix} 2+2i \ 1 \end{pmatrix} \):\[3 \begin{pmatrix} 2+2i \ 1 \end{pmatrix} = \begin{pmatrix} 6 + 6i \ 3 \end{pmatrix},\]and thus \( \lambda = 6+6i \). Therefore, \( \mathbf{K}_3 \) is not an eigenvector.

Key Concepts

Eigenvalues

In linear algebra, an eigenvalue is a special scalar associated with a linear transformation represented by a matrix. When a matrix \( \mathbf{A} \) transforms an eigenvector \( \mathbf{v} \), the transformation results in a scalar multiple of the original vector. This scalar is known as the eigenvalue, denoted as \( \lambda \). Analyzing eigenvalues allows us to understand how matrices stretch or shrink vectors without altering their directions, except possibly reversing them. Here are key points to grasp:

• Eigenvalues can be real or complex numbers.
• Eigenvalues provide insight into the system's stability, vibrations, and other properties, depending on the context of the problem.
• To find an eigenvalue, we solve the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix.

Understanding eigenvalues helps us comprehend the behaviors of different transformations represented by matrices.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, and linear transformations. It provides the foundation for modern computational methods and has wide applications in engineering, physics, computer science, and more. Some fundamental concepts include:

• Understanding vector spaces, where vectors are entities that have both magnitude and direction.
• Exploring linear transformations, which are mappings that preserve vector addition and scalar multiplication.
• Utilizing operations with matrices and determinants to solve systems of linear equations.

The role of eigenvectors and eigenvalues is paramount in linear algebra since they simplify complex matrix operations and help in understanding the intrinsic properties of matrices.

Matrix Algebra

Matrix algebra deals with operations concerning matrices, which are rectangular arrays of numbers, symbols, or expressions. Here, operations like matrix addition, subtraction, multiplication, and inversion come into play. Among these, understanding

• Matrix multiplication, where the elements of rows are multiplied and summed with columns.
• Determinants, which are scalar values that can indicate whether a matrix has an inverse.
• Matrix inversion, which finds another matrix that, when multiplied with the original, yields the identity matrix.

is crucial. Matrix algebra forms the crux of computational operations in many algorithms involving data transformations, linear mappings, and solving systems of mathematical equations. Calculating eigenvalues and eigenvectors requires a good grasp of matrix manipulations and understanding their roles in these calculations.