Problem 4 - Advanced Engineering Mathematics

Question

In Problems 1-6, 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

\( \mathbf{K}_2 \) is an eigenvector with eigenvalue \( 2i \). \( \mathbf{K}_1 \) and \( \mathbf{K}_3 \) are not eigenvectors.

Step 1: Defining Eigenvectors and Eigenvalues

Eigenvectors of a matrix \( \mathbf{A} \) are the non-zero vectors \( \mathbf{v} \) such that \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \), where \( \lambda \) is the corresponding eigenvalue. A zero vector cannot be an eigenvector.

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

Since \( \mathbf{K}_1 = \begin{pmatrix} 0 \ 0 \end{pmatrix} \) is a zero vector, it cannot be an eigenvector of any matrix. Thus, \( \mathbf{K}_1 \) is not an eigenvector of \( \mathbf{A} \).

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

Let's calculate \( \mathbf{A} \mathbf{K}_2 \) to verify if it yields a scalar multiple of \( \mathbf{K}_2 \):\[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 2 & 8 \ -1 & -2 \end{pmatrix} \begin{pmatrix} 2 + 2i \ -1 \end{pmatrix} = \begin{pmatrix} 2(2 + 2i) + 8(-1) \ -1(2 + 2i) - 2(-1) \end{pmatrix} = \begin{pmatrix} 4 + 4i - 8 \ -2 - 2i + 2 \end{pmatrix} \]Simplifying gives:\[\begin{pmatrix} -4 + 4i \ -2i \end{pmatrix}\]Compare it with \( \mathbf{K}_2 \).

Step 4: Determine if \(\mathbf{K}_2\) is an Eigenvector

We need to find \( \lambda \) such that \( (-4 + 4i) = \lambda (2 + 2i) \) and \( -2i = \lambda(-1) \), which implies \( \lambda = 2i \). This shows that \( \mathbf{K}_2 \) is an eigenvector with eigenvalue \( \lambda = 2i \).

Step 5: Compute \( \mathbf{A} \mathbf{K}_3 \)

Calculate \( \mathbf{A} \mathbf{K}_3 \):\[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 2 & 8 \ -1 & -2 \end{pmatrix} \begin{pmatrix} 2 + 2i \ 1 \end{pmatrix} = \begin{pmatrix} 2(2 + 2i) + 8(1) \ -1(2 + 2i) - 2(1) \end{pmatrix} = \begin{pmatrix} 4 + 4i + 8 \ -2 - 2i - 2 \end{pmatrix} \]Simplifying gives:\[\begin{pmatrix} 12 + 4i \ -4 - 2i \end{pmatrix}\]Compare it with \( \mathbf{K}_3 \).

Step 6: Determine if \(\mathbf{K}_3\) is an Eigenvector

The calculation \( 12 + 4i eq \lambda(2 + 2i) \) and \( -4 - 2i eq \lambda(1) \) do not simultaneously yield a consistent \( \lambda \). Therefore, \( \mathbf{K}_3 \) is not an eigenvector of \( \mathbf{A} \).

Key Concepts

Linear Algebra

Linear algebra is a crucial branch of mathematics that deals with vector spaces and linear mappings between these spaces. It forms the foundation for many other areas of mathematics and is essential for solving various real-world problems. In linear algebra, we deal with concepts such as vectors, matrices, and linear transformations. These concepts help to understand how different dimensions relate to each other. An important aspect of linear algebra is the study of eigenvectors and eigenvalues. Eigenvectors are special vectors that, when transformed by a matrix, result only in a scaled version of themselves. The scalar multiplier is known as the eigenvalue. This relationship is fundamental in many fields, including physics, computer science, and economics, because it provides insights into system behavior and stability. Understanding linear algebra allows us to represent complex data sets more simply and efficiently.

Matrix Operations

Matrix operations are a set of algebraic computations involving matrices, which are rectangular arrays of numbers. These operations include addition, subtraction, multiplication, and finding inverses. Matrix multiplication is particularly significant because it forms the basis of many linear transformations. It involves calculating the dot product of rows and columns to produce a new matrix. When working with eigenvectors and eigenvalues, matrix multiplication is frequently used. The equation \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \) describes the operation where matrix \( \mathbf{A} \) is multiplied by vector \( \mathbf{v} \), resulting in a scaled vector \( \lambda \mathbf{v} \). This computation helps verify if a particular vector is an eigenvector of a matrix and identify the corresponding eigenvalue. Understanding these operations is crucial for solving systems of linear equations, performing data transformations, and modeling various phenomena in engineering and science.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. They are usually expressed in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary component is denoted by \( i \), which is defined as the square root of \( -1 \). Complex numbers extend the idea of one-dimensional real numbers to a two-dimensional complex plane, creating a broader scope of mathematical operations.In the context of eigenvectors, complex numbers often appear when the matrix has non-real eigenvalues. These complex eigenvalues imply oscillatory behavior, which is common in systems confronted with periodic processes, like alternating currents in electrical engineering. Understanding how to work with complex numbers, including how to perform arithmetic operations and how to graph them on the complex plane, is important for comprehending more intricate linear systems that cannot be described solely with real numbers.