Problem 5 - 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. $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 2 \\ -2 & 1 & -2 \\ 2 & 2 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right), $$ $$ \mathbf{K}_{2}=\left(\begin{array}{r} 4 \\ -4 \\ 0 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right) $$

Step-by-Step Solution

Answer

None of the vectors are eigenvectors of \(\mathbf{A}\).

Step 1: Understand Eigenvector and Matrix Equation

An eigenvector of a matrix \(\mathbf{A}\) is a non-zero vector \(\mathbf{v}\) such that \(\mathbf{A}\mathbf{v} = \lambda\mathbf{v}\), where \(\lambda\) is called the eigenvalue. To determine if a vector is an eigenvector of \(\mathbf{A}\), you multiply the matrix by the vector and check if the result is a scalar multiple of the original vector.

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

Calculate the product \(\mathbf{A} \mathbf{K}_1\): \[\mathbf{A} \mathbf{K}_1 = \begin{pmatrix} 1 & -2 & 2 \ -2 & 1 & -2 \ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 0 \end{pmatrix} + \begin{pmatrix} -2 \ 1 \ 2 \end{pmatrix} + \begin{pmatrix} 2 \ -2 \ 1 \end{pmatrix} = \begin{pmatrix} 0 + (-2) + 2 \ 0 + 1 - 2 \ 0 + 2 + 1 \end{pmatrix} = \begin{pmatrix} 0 \ -1 \ 3 \end{pmatrix}. \] Since \(\begin{pmatrix} 0 \ -1 \ 3 \end{pmatrix}\) is not a scalar multiple of \(\begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}\), \( \mathbf{K}_1 \) is not an eigenvector.

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

Calculate the product \(\mathbf{A} \mathbf{K}_2\): \[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} 1 & -2 & 2 \ -2 & 1 & -2 \ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} 4 \ -4 \ 0 \end{pmatrix} = \begin{pmatrix} 4 \ -8 \ 8 \end{pmatrix} + \begin{pmatrix} 8 \ -4 \ 0 \end{pmatrix} + \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} = \begin{pmatrix} 12 \ -12 \ 8 \end{pmatrix}. \] The result \(\begin{pmatrix} 12 \ -12 \ 8 \end{pmatrix}\) is not a scalar multiple of \(\begin{pmatrix} 4 \ -4 \ 0 \end{pmatrix}\), so \(\mathbf{K}_2\) is not an eigenvector.

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

Calculate the product \(\mathbf{A} \mathbf{K}_3\): \[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} 1 & -2 & 2 \ -2 & 1 & -2 \ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} -1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} -1 \ 2 \ 2 \end{pmatrix} + \begin{pmatrix} 2 \ 1 \ 2 \end{pmatrix} + \begin{pmatrix} 2 \ -2 \ 1 \end{pmatrix} = \begin{pmatrix} -1 + 2 + 2 \ 2 + 1 - 2 \ 2 + 2 + 1 \end{pmatrix} = \begin{pmatrix} 3 \ 1 \ 5 \end{pmatrix}. \] Since \(\begin{pmatrix} 3 \ 1 \ 5 \end{pmatrix}\) is not a scalar multiple of \(\begin{pmatrix} -1 \ 1 \ 1 \end{pmatrix}\), \(\mathbf{K}_3\) is not an eigenvector.

Step 5: Conclusion

None of the vectors \(\mathbf{K}_1\), \(\mathbf{K}_2\), or \(\mathbf{K}_3\) are eigenvectors of matrix \(\mathbf{A}\), as none satisfy the condition \(\mathbf{A}\mathbf{v} = \lambda\mathbf{v}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying rows of the first matrix by columns of the second, summing the products to get a single number—a part of the resulting matrix. This is a systematic way to derive a new matrix that combines the two matrices. For a matrix \( A \) and vector \( v \), matrix multiplication is used to see how the transformation described by \( A \) affects \( v \).

• To perform matrix multiplication, ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• The resulting product will be a matrix with dimensions dictated by the rows of the first matrix and columns of the second.

This operation is essential for problems involving eigenvectors, as it helps check if vector transformations lead to scalar multiples.

Eigenvalues

Eigenvalues are crucial in understanding eigenvectors and matrix transformations. When a vector is an eigenvector of a matrix, it is stretched by a factor known as an eigenvalue when multiplied by the matrix. This means that the direction remains unchanged, but the magnitude is scaled.

• If \( \mathbf{A}\mathbf{v} = \lambda\mathbf{v} \), then \( \lambda \) is the eigenvalue corresponding to the eigenvector \( \mathbf{v} \).
• Not every vector has an eigenvalue; only eigenvectors with specific orientations do.

In solving problems, you'll often determine if vectors are eigenvectors of matrices by checking the resulting vector after multiplication for alignment with the original vector's direction.

Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors and matrices, essential for understanding the nature of space and transformations within that space. It provides tools to model and solve sub-problems in larger frameworks, notably via vector spaces and matrix operations.

• It handles vector addition, scalar multiplication, and more complex operations like matrix multiplication and determinant calculation.
• Linear algebra is crucial in fields like computer science, physics, and engineering, providing a framework to deal with multi-dimensional vector spaces.

Working with linear algebra typically involves understanding how mathematical structures can describe transformations and solutions to systems of linear equations.

Vector Spaces

A vector space, or linear space, is a collection of vectors where vector addition and scalar multiplication are defined. This concept allows mathematicians and scientists to work within a structured framework of rules describing relationships between vectors.

• Vectors in a vector space can be added together and multiplied by scalars to produce another vector within the same space.
• Vector spaces can be of any dimension, with common examples being two-dimensional and three-dimensional spaces.

Understanding vector spaces is crucial to comprehend other concepts in linear algebra, such as eigenvalues and eigenvectors, because these exist within and interact according to the rules governing these spaces.