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

Step-by-Step Solution

Answer

Eigenvectors: \( \mathbf{K}_2 \) with eigenvalue 1; \( \mathbf{K}_3 \) with eigenvalue 3.

Step 1: Define an Eigenvector

An eigenvector of a matrix \( \mathbf{A} \) is a nonzero vector \( \mathbf{v} \) such that \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \), where \( \lambda \) is the eigenvalue.

Step 2: Check First Vector

Calculate \( \mathbf{A} \mathbf{K}_1 \) and see if it is a scalar multiple of \( \mathbf{K}_1 \).Calculate:\[ \mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 2 & -1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 1 \ 2-\sqrt{2} \end{pmatrix} = \begin{pmatrix} 2 - (2-\sqrt{2}) \ 2 - 2(2-\sqrt{2}) \end{pmatrix} = \begin{pmatrix} \sqrt{2} \ -2+2\sqrt{2} \end{pmatrix} \]Since the resulting vector \( \begin{pmatrix} \sqrt{2} \ -2+2\sqrt{2} \end{pmatrix} \) is not a scalar multiple of \( \mathbf{K}_1 \), \( \mathbf{K}_1 \) is not an eigenvector.

Step 3: Check Second Vector

Calculate \( \mathbf{A} \mathbf{K}_2 \) and determine if it is a scalar multiple of \( \mathbf{K}_2 \).Calculate:\[ \mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 2 & -1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 2+\sqrt{2} \ 2 \end{pmatrix} = \begin{pmatrix} 2(2+\sqrt{2}) - 2 \ 2(2+\sqrt{2}) - 4 \end{pmatrix} = \begin{pmatrix} 4 + 2\sqrt{2} - 2 \ 4 + 2\sqrt{2} - 4 \end{pmatrix} = \begin{pmatrix} 2 + 2\sqrt{2} \ 2\sqrt{2} \end{pmatrix} \]This result is a scalar multiple of \( \mathbf{K}_2 \) with a factor of 1, indicating that \( \mathbf{K}_2 \) is an eigenvector with eigenvalue 1.

Step 4: Check Third Vector

Calculate \( \mathbf{A} \mathbf{K}_3 \) and determine if it is a scalar multiple of \( \mathbf{K}_3 \).Calculate:\[ \mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 2 & -1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} \sqrt{2} \ -\sqrt{2} \end{pmatrix} = \begin{pmatrix} 2\sqrt{2} + \sqrt{2} \ 2\sqrt{2} + \sqrt{2} \end{pmatrix} = \begin{pmatrix} 3\sqrt{2} \ 3\sqrt{2} \end{pmatrix} \] This result is a scalar multiple of \( \mathbf{K}_3 \) with a factor of 3, indicating that \( \mathbf{K}_3 \) is an eigenvector with eigenvalue 3.

Key Concepts

Eigenvalues

Eigenvalues are a cornerstone of linear algebra, acting as special scalars associated with a square matrix and its eigenvectors. When you multiply a matrix \( \mathbf{A} \) by one of its eigenvectors \( \mathbf{v} \), the result is simply the eigenvector \( \mathbf{v} \) scaled by a factor, which is the eigenvalue \( \lambda \). This relationship is expressed as \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \).
Understanding eigenvalues can provide valuable insights into many linear transformations, such as rotations, scalings, and other manipulations of space.

• They help determine the stability and dynamics of systems in differential equations.
• They are used in principal component analysis (PCA) for dimensionality reduction in data science.

Calculating eigenvalues typically involves solving the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix of the same dimensions as \( \mathbf{A} \). Solving this equation yields the eigenvalues \( \lambda \) that correspond to the critical values of the transformation represented by \( \mathbf{A} \).

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra, heavily relied upon when dealing with eigenvectors and eigenvalues. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Then, each element in the resultant matrix is computed as the sum of products between corresponding elements of the row from the first matrix and the column from the second matrix.
This operation is not only used to determine if a vector is an eigenvector of a given matrix, but also has broader applications:

• Linear transformations: Implement transformations in space, such as rotations or scaling.
• System of equations: Represent and solve multiple linear equations simultaneously.
• Computer graphics: Handle transformations for rendering 3D models on a 2D screen.

When multiplying a matrix by a vector, such as in chapters discussing eigenvalues and eigenvectors, each entry of the vector is treated as a column matrix, resulting in another vector.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It involves the study of lines, planes, and subspaces, but foremostly matrices and vectors. Linear algebra forms the theoretical underpinning of many areas in mathematics and applied sciences.
Fundamental concepts of linear algebra include:

• Vector spaces: Collections of vectors that can be added together and multiplied by scalars.
• Linear transformations: Functions that map vectors to vectors in a linear manner, preserving vector addition and scalar multiplication.
• Systems of linear equations: Sets of equations that can be represented and solved using matrix methods.

Understanding these concepts provides the foundation for exploring more complex topics like eigenvectors and eigenvalues, matrix decompositions, and solving optimization problems. Linear algebra's utility makes it indispensable in fields such as physics, computer science, data analysis, and more.