Problem 1 - 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. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} 4 & 2 \\ 5 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 5 \\ -2 \end{array}\right), \mathbf{K}_{2}=\left(\begin{array}{l} 2 \\ 5 \end{array}\right), \\ &\mathbf{K}_{3}=\left(\begin{array}{r} -2 \\ 5 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

\( \mathbf{K}_3 \) is an eigenvector with eigenvalue \( -1 \).

Step 1: Understanding Eigenvectors and Eigenvalues

An eigenvector of a matrix \( \mathbf{A} \) is a vector \( \mathbf{v} \) that satisfies the equation \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \), where \( \lambda \) is the eigenvalue. We need to check which of the given vectors is an eigenvector of the matrix \( \mathbf{A} \).

Step 2: Checking for \( \mathbf{K}_1 \)

Multiply matrix \( \mathbf{A} \) by \( \mathbf{K}_1 = \begin{pmatrix} 5 \ -2 \end{pmatrix} \). Calculate: \[\mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 4 & 2 \ 5 & 1 \end{pmatrix} \begin{pmatrix} 5 \ -2 \end{pmatrix} = \begin{pmatrix} (4)(5) + (2)(-2) \ (5)(5) + (1)(-2) \end{pmatrix} = \begin{pmatrix} 16 \ 23 \end{pmatrix}.\]This is not a scalar multiple of \( \mathbf{K}_1 \), so \( \mathbf{K}_1 \) is not an eigenvector.

Step 3: Checking for \( \mathbf{K}_2 \)

Multiply matrix \( \mathbf{A} \) by \( \mathbf{K}_2 = \begin{pmatrix} 2 \ 5 \end{pmatrix} \). Calculate: \[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 4 & 2 \ 5 & 1 \end{pmatrix} \begin{pmatrix} 2 \ 5 \end{pmatrix} = \begin{pmatrix} (4)(2) + (2)(5) \ (5)(2) + (1)(5) \end{pmatrix} = \begin{pmatrix} 18 \ 15 \end{pmatrix}.\]\( \begin{pmatrix} 18 \ 15 \end{pmatrix} = 3 \begin{pmatrix} 6 \ 5 \end{pmatrix} = 3 \mathbf{K}_2 \), so \( \mathbf{K}_2 \) is not an eigenvector because it cannot be written as a direct multiple of \( \mathbf{K}_2 \).

Step 4: Checking for \( \mathbf{K}_3 \)

Multiply matrix \( \mathbf{A} \) by \( \mathbf{K}_3 = \begin{pmatrix} -2 \ 5 \end{pmatrix} \). Calculate: \[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 4 & 2 \ 5 & 1 \end{pmatrix} \begin{pmatrix} -2 \ 5 \end{pmatrix} = \begin{pmatrix} (4)(-2) + (2)(5) \ (5)(-2) + (1)(5) \end{pmatrix} = \begin{pmatrix} 2 \ -5 \end{pmatrix}.\]\( \begin{pmatrix} 2 \ -5 \end{pmatrix} = -1 \begin{pmatrix} -2 \ 5 \end{pmatrix} = -1 \mathbf{K}_3 \). Therefore, \( \mathbf{K}_3 \) is an eigenvector with eigenvalue \( -1 \).

Key Concepts

Eigenvalues

Understanding eigenvalues is crucial when diving into linear algebra concepts. An eigenvalue \( \lambda \) is a special scalar that, when a matrix \( \mathbf{A} \) is multiplied by one of its eigenvectors \( \mathbf{v} \), results in the eigenvector being scaled by \( \lambda \). This relationship is elegantly captured in the equation \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \).

To find eigenvalues, one typically needs to solve the characteristic equation \( \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix of the same dimension as \( \mathbf{A} \). Solving this equation can involve finding the roots of a polynomial, which provides the eigenvalues.

In our original problem, the matrix \( \mathbf{A} \) turned the vector \( \mathbf{K}_3 \) into a scalar multiple of itself by \( -1 \), confirming \( -1 \) as an eigenvalue. Understanding which combinations of \( \mathbf{A} \) and \( \mathbf{v} \) return a scalar multiple of \( \mathbf{v} \) can reveal much about their structural properties, such as stretching, scaling, or directional transformations.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves the dot product of rows and columns from two matrices. Here’s how it works: each element of the resulting matrix is the sum of products of corresponding elements from rows and columns of the two matrices being multiplied. Let's look at a simple step-by-step:
- Take a row from the first matrix and a column from the second matrix.
- Multiply elements in the row by elements in the column one by one, then sum all the products.
- Place this summed product in the resulting matrix.
- Repeat for every row and column pair.

In our exercise scenario, the matrix \( \mathbf{A} \) is multiplied by each candidate eigenvector to check for eigenvector property, which reveals whether the product yields a scalar multiple of the original vector. For example, when multiplying \( \mathbf{A} \) with \( \mathbf{K}_3 \), the use of matrix multiplication helps confirm that the operation results in \( \begin{pmatrix} 2 \ -5 \end{pmatrix} = -1 \mathbf{K}_3 \), demonstrating the validity of \( \mathbf{K}_3 \) as an eigenvector.

Linear Algebra

Linear algebra is the cornerstone for solving many problems in mathematics and engineering. It encompasses concepts like vectors, matrices, and spaces, making it instrumental in modeling and solving real-world situations. One of its powerful tools is the ability to work with transformations using matrices, understanding them through eigenvectors and eigenvalues.

Key components to grasp include:
- **Vectors and Spaces:** Basic elements that represent directions and magnitudes. They serve as building blocks for higher-dimensional analysis.
- **Matrices:** Arrays of numbers representing linear transformations. They allow operations like rotations, scalings, or translations in spaces.
- **Determinants and Eigenvalues:** Determinants help understand matrix properties, like invertibility and volume scaling, while eigenvalues describe the intrinsic transformations a matrix imposes.

Understanding these essentials in linear algebra helps in leveraging this powerful field to solve complex problems, from graphics in computer science to economic models. Recognizing how eigenvectors and eigenvalues dictate matrix behavior is a fundamental skill, greatly enhancing comprehension and application of linear transformations.