Problem 2 - 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. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{cc} 2 & -1 \\ 2 & -2 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{c} 1 \\ 2-\sqrt{2} \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{c} 2+\sqrt{2} \\ 2 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{r} \sqrt{2} \\ -\sqrt{2} \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

Only \( \mathbf{K}_3 \) is an eigenvector with eigenvalue \( \lambda = 2 \).

Step 1: Understand Eigenvector and Eigenvalue Definitions

An eigenvector of a matrix \( \mathbf{A} \) is a non-zero vector \( \mathbf{x} \) such that \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \), where \( \lambda \) is a scalar called the eigenvalue. The problem asks to check each given vector to see if it satisfies \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \) for some \( \lambda \).

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

Compute \( \mathbf{A} \mathbf{K}_1 \):\[ \mathbf{A} \begin{pmatrix} 1 \ 2-\sqrt{2} \end{pmatrix} = \begin{pmatrix} 2 & -1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 1 \ 2-\sqrt{2} \end{pmatrix} = \begin{pmatrix} 4 - \sqrt{2} \ 2 - 2\sqrt{2} \end{pmatrix} \]Check if this result is a scalar multiple of \( \mathbf{K}_1 \). Solve \( \lambda \begin{pmatrix} 1 \ 2-\sqrt{2} \end{pmatrix} = \begin{pmatrix} 4 - \sqrt{2} \ 2 - 2\sqrt{2} \end{pmatrix} \), which does not hold. Hence, \( \mathbf{K}_1 \) is not an eigenvector.

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

Compute \( \mathbf{A} \mathbf{K}_2 \):\[ \mathbf{A} \begin{pmatrix} 2+\sqrt{2} \ 2 \end{pmatrix} = \begin{pmatrix} 2 & -1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 2+\sqrt{2} \ 2 \end{pmatrix} = \begin{pmatrix} 2\sqrt{2} \end{pmatrix} \]Check if \( \mathbf{K}_2 \) is a scalar multiple of \( \mathbf{A} \mathbf{K}_2 \). Solve \( \lambda \begin{pmatrix} 2+\sqrt{2} \ 2 \end{pmatrix} = \begin{pmatrix} 2\sqrt{2} \ 0 \end{pmatrix} \), which does not hold. \( \mathbf{K}_2 \) is not an eigenvector.

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

Compute \( \mathbf{A} \mathbf{K}_3 \):\[ \mathbf{A} \begin{pmatrix} \sqrt{2} \ -\sqrt{2} \end{pmatrix} = \begin{pmatrix} \sqrt{2} + \sqrt{2} \ 0 \end{pmatrix} = \begin{pmatrix} 2\sqrt{2} \ 0 \end{pmatrix} \]Check if this is a scalar multiple of \( \mathbf{K}_3 \). We find \( 2 \begin{pmatrix} \sqrt{2} \ -\sqrt{2} \end{pmatrix} = \begin{pmatrix} 2\sqrt{2} \ -2\sqrt{2} \end{pmatrix} \), which implies \( \mathbf{A} \mathbf{K}_3 = 2 \mathbf{K}_3 \). Hence, \( \mathbf{K}_3 \) is an eigenvector associated with eigenvalue \( \lambda = 2 \).

Step 5: Conclude which vector is an eigenvector

From the evaluations, only \( \mathbf{K}_3 \) is an eigenvector of \( \mathbf{A} \) with the eigenvalue \( \lambda = 2 \). \( \mathbf{K}_1 \) and \( \mathbf{K}_2 \) are not eigenvectors of the given matrix.

Key Concepts

Matrix Analysis

Matrix analysis is a critical area within mathematics, dealing particularly with the study of matrices and matrix operations. It explores how matrices behave when subject to various operations such as multiplication, addition, or transformation. An essential aspect of matrix analysis is understanding how matrices interact with vectors.

Consider when a matrix is multiplied by a vector. The result can either stretch, shrink, or reverse the direction of the vector. This operation is key to numerous applications, including systems of linear equations, computer graphics, or even in the understanding and simulation of dynamic systems.

In our given exercise, matrix \( \mathbf{A} \) is analyzed to determine which vectors are its eigenvectors. The process involves multiplying \( \mathbf{A} \) by candidate vectors and checking for scalar multiples. When a matrix transformation results in a vector that is merely a scaled version of the original, it indicates the presence of an eigenvector.

Eigenvalues

Eigenvalues, represented often by the symbol \( \lambda \), are scalars associated with matrices that signify how eigenvectors are transformed under a matrix operation. When a vector \( \mathbf{x} \) is an eigenvector of a matrix \( \mathbf{A} \), multiplying \( \mathbf{x} \) by \( \mathbf{A} \) results in the vector being scaled by \( \lambda \). Mathematically, this is represented by the equation \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \).

Determining an eigenvalue for a matrix involves solving the above equation for \( \lambda \). In our exercise, the matrix \( \mathbf{A} \) was combined with various vectors to check for these specific multipliers. If for a given vector the resulting transformed vector is a scaled version of the original, then that scale factor is the eigenvalue.

For instance, in this exercise, by evaluating \( \mathbf{K}_3 \), it is established that \( \mathbf{A} \mathbf{K}_3 = 2 \mathbf{K}_3 \). Thus, \( \lambda = 2 \) is the eigenvalue for \( \mathbf{K}_3 \), confirming it is an eigenvector for matrix \( \mathbf{A} \).

Linear Algebra

Linear algebra is the study of vectors, vector spaces, linear transformations, and systems of linear equations. It provides the tools necessary for working with matrices and solving linear systems, which are prevalent in fields such as physics, engineering, computer science, and statistics.

A fundamental concept in linear algebra that relates to the given problem is the identification of the relationship between matrices and vectors. Particularly, identifying eigenvectors and eigenvalues aids in understanding matrix behaviors. Eigenvectors remain invariant in direction under the application of a matrix, being scaled by the eigenvalue.

The exercise highlights the concepts of eigenvectors and eigenvalues in practical applications of linear algebra. It tests vectors against a matrix \( \mathbf{A} \) to determine if they maintain their direction post-multiplication, hence verifying the theory behind eigenvalues and eigenvectors directly. It's an excellent demonstration of linear algebra concepts applied to matrix analysis. This study is not only theoretical but has numerous real-world applications in optimizing systems and predicting dynamic processes.