Problem 6 - 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. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} -1 \\ 4 \\ 3 \end{array}\right) \\ &\mathbf{K}_{2}=\left(\begin{array}{l} 1 \\ 4 \\ 3 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{l} 3 \\ 1 \\ 4 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

None of \(\mathbf{K}_1, \mathbf{K}_2, \mathbf{K}_3\) are eigenvectors of \(\mathbf{A}\).

Step 1: Understand the problem

We need to check if the given vectors \(\mathbf{K}_1, \mathbf{K}_2,\) and \(\mathbf{K}_3\) are eigenvectors of the matrix \(\mathbf{A}\). An eigenvector \(\mathbf{v}\) of a matrix \(\mathbf{A}\) satisfies the equation \(\mathbf{A}\mathbf{v} = \lambda \mathbf{v}\), where \(\lambda\) is the eigenvalue.

Step 2: Check \(\mathbf{K}_1\) as eigenvector

Compute \(\mathbf{A}\mathbf{K}_1\) and check if it is a scalar multiple of \(\mathbf{K}_1\).\[\mathbf{A} \mathbf{K}_1 = \begin{pmatrix} -1 & 1 & 0 \ 1 & 2 & 1 \ 0 & 3 & -1 \end{pmatrix}\begin{pmatrix} -1 \ 4 \ 3 \end{pmatrix} = \begin{pmatrix} 5 \ 11 \ 9 \end{pmatrix}\]This is not a scalar multiple of \(\begin{pmatrix} -1 \ 4 \ 3 \end{pmatrix}\), so \(\mathbf{K}_1\) is not an eigenvector.

Step 3: Check \(\mathbf{K}_2\) as eigenvector

Compute \(\mathbf{A}\mathbf{K}_2\) and check if it is a scalar multiple of \(\mathbf{K}_2\).\[\mathbf{A} \mathbf{K}_2 = \begin{pmatrix} -1 & 1 & 0 \ 1 & 2 & 1 \ 0 & 3 & -1 \end{pmatrix}\begin{pmatrix} 1 \ 4 \ 3 \end{pmatrix} = \begin{pmatrix} 3 \ 11 \ 9 \end{pmatrix}\]This is not a scalar multiple of \(\begin{pmatrix} 1 \ 4 \ 3 \end{pmatrix}\), so \(\mathbf{K}_2\) is not an eigenvector.

Step 4: Check \(\mathbf{K}_3\) as eigenvector

Compute \(\mathbf{A}\mathbf{K}_3\) and check if it is a scalar multiple of \(\mathbf{K}_3\).\[\mathbf{A} \mathbf{K}_3 = \begin{pmatrix} -1 & 1 & 0 \ 1 & 2 & 1 \ 0 & 3 & -1 \end{pmatrix}\begin{pmatrix} 3 \ 1 \ 4 \end{pmatrix} = \begin{pmatrix} -2 \ 10 \ -1 \end{pmatrix}\]This is not a scalar multiple of \(\begin{pmatrix} 3 \ 1 \ 4 \end{pmatrix}\), so \(\mathbf{K}_3\) is not an eigenvector.

Key Concepts

Eigenvalues

Eigenvalues are a fundamental concept in linear algebra. They are associated with a square matrix and provide important information about the matrix's properties. Simply speaking, an eigenvalue, denoted as \( \lambda \), is a scalar that satisfies the equation \( \mathbf{A}\mathbf{v} = \lambda \mathbf{v} \), where \( \mathbf{A} \) is a matrix and \( \mathbf{v} \) is the eigenvector. This equation means that when the matrix \( \mathbf{A} \) multiplies the eigenvector \( \mathbf{v} \), the result is the eigenvector scaled by the eigenvalue.

To find eigenvalues, one typically solves the characteristic equation: \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix of the same size as \( \mathbf{A} \). Solving this determinant equation yields the possible eigenvalues of the matrix \( \mathbf{A} \).
Understanding eigenvalues helps in various applications such as stability analysis, vibration analysis, and quantum mechanics. It provides insight into whether a system will amplify signals, reach equilibrium, or act in some periodic manner.

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra and involves taking two matrices and producing another matrix. For matrices to multiply, the number of columns in the first matrix must equal the number of rows in the second matrix.

Consider two matrices \( \mathbf{A} \) and \( \mathbf{B} \). The element at the position \( i,j \) in the resulting matrix \( \mathbf{C} = \mathbf{A} \mathbf{B} \) is given by the dot product of the \( i \)-th row of \( \mathbf{A} \) and the \( j \)-th column of \( \mathbf{B} \). This operation is not commutative, which means that \( \mathbf{A} \mathbf{B} eq \mathbf{B} \mathbf{A} \) in general.

Matrix multiplication is essential in finding eigenvectors and eigenvalues because it allows us to perform operations like \( \mathbf{A}\mathbf{v} \) to check if \( \mathbf{v} \) is indeed an eigenvector of \( \mathbf{A} \). It also plays a crucial role in transformations, computer graphics, and solving systems of linear equations.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It provides methods to solve systems of linear equations, transform geometric figures, and much more.

Core concepts in linear algebra include vector spaces, where vectors are organized, and operations are defined. Matrices represent linear transformations, which are essentially functions that map vectors onto other vectors in a predictable way. Eigenvectors and eigenvalues, as discussed, are part of these transformations, describing how certain vectors only scale and do not rotate or otherwise transform.

In practical applications, linear algebra is used in fields like computer science for algorithms, physics for modeling physical systems, engineering for structural analysis, and more. It provides the tools necessary to understand and manipulate various mathematical systems.