Question

A matrix \(A\) and one of its eigenvalues are given. Find an eigenvector of A for the given eigenvalue. $$ \begin{array}{l} A=\left[\begin{array}{cc} 16 & 6 \\ -18 & -5 \end{array}\right] \\ \lambda=4 \end{array} $$

Short answer

The eigenvector is \( \begin{bmatrix} 1 \\ -2 \end{bmatrix} \).

Step-by-step solution

Understand the Eigenvector Equation

The eigenvector equation is given by \( A\mathbf{v} = \lambda \mathbf{v} \), where \( A \) is a matrix, \( \lambda \) is an eigenvalue, and \( \mathbf{v} \) is the corresponding eigenvector. We need to solve \( A\mathbf{v} = 4\mathbf{v} \) for \( \mathbf{v} \).

Set Up the Homogeneous System of Equations

Rearrange the equation to \( (A - 4I)\mathbf{v} = \mathbf{0} \), where \( I \) is the identity matrix. First, find \( A - 4I \): \[ A - 4I = \begin{bmatrix} 16 & 6 \ -18 & -5 \end{bmatrix} - \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} = \begin{bmatrix} 12 & 6 \ -18 & -9 \end{bmatrix}. \]

Solve the System of Linear Equations

The system \( \begin{bmatrix} 12 & 6 \ -18 & -9 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \) must be solved. The equations are \( 12x + 6y = 0 \) and \( -18x - 9y = 0 \). Simplify them: divide the first by 6 and the second by -9, resulting in \( 2x + y = 0 \) and \( 2x + y = 0 \). They are identical, so \( y = -2x \).

Find the Eigenvector

Choose a simple solution for \( x \). Let \( x = 1 \), then \( y = -2(1) = -2 \). An eigenvector of \( A \) corresponding to the eigenvalue 4 is \( \begin{bmatrix} 1 \ -2 \end{bmatrix} \). Check the result by substituting back: \( \begin{bmatrix} 16 & 6 \ -18 & -5 \end{bmatrix} \begin{bmatrix} 1 \ -2 \end{bmatrix} = \begin{bmatrix} 4 \ -8 \end{bmatrix} = 4 \begin{bmatrix} 1 \ -2 \end{bmatrix} \).

Key concepts

Eigenvalues

Eigenvalues are special numbers associated with a square matrix in linear algebra. These numbers provide insight into a matrix's characteristics and its transformational properties. To find an eigenvalue for a given matrix, one needs to solve the characteristic equation, which is obtained from the determinant equation \(\det(A - \lambda I) = 0\). Here, \(A\) is the matrix in question, \(\lambda\) represents the eigenvalue, and \(I\) is the identity matrix of the same size as \(A\). Therefore, eigenvalues tell us how much a matrix can be "stretched" or "compressed" along its corresponding eigenvectors. For example, if a matrix has an eigenvalue of 4, this indicates that the matrix can scale a vector (its eigenvector) by a factor of 4. This property is crucial in understanding systems of differential equations and stability analysis. When working on practical problems, students should be aware that different matrices will have varying numbers of eigenvalues, some of which may be complex numbers.

Matrix Algebra

Matrix Algebra involves various operations on matrices, including addition, subtraction, multiplication, and finding inverses. A fundamental aspect of matrix algebra is the manipulation of matrices to reveal their properties and solve matrix equations. One important operation in matrix algebra is the calculation of eigenvectors and eigenvalues, vital for numerous applications in science and engineering.For instance, in the context of finding an eigenvector with a given eigenvalue, matrix algebra is used to adjust the matrix so that standard linear solving techniques can apply. Consider the example where you have a matrix \(A\) and need to find an eigenvector corresponding to an eigenvalue \(\lambda\). This involves computing \(A - \lambda I\), transforming the problem into a system of linear equations.Matrix algebra serves as a bridge between abstract mathematical concepts and applied problem-solving in fields like physics and computer science. Understanding matrix operations helps in solving complex problems in linear demensions, making matrix algebra an indispensable tool.

Linear Equations

Linear Equations form the backbone of linear algebra, representing equations of lines or planes in mathematical space. Systems of linear equations can often be found when working with matrices, especially in the context of eigenvectors and eigenvalues. A system of linear equations is typically expressed in matrix form, and solving this system is key to finding eigenvectors. The process involves setting up equations based on altered matrices, such as solving \((A - \lambda I)\mathbf{v} = \mathbf{0}\). For example, for a matrix \(A\) and an eigenvalue \(\lambda\), we rearrange to find the matrix \(A - \lambda I\), then solve the resulting homogeneous linear equations.In many applications, these equations reveal fundamental properties about a system, like stability and responses to external forces. Linear equations are straightforward to manipulate mathematically, which makes them ideal for numerical solutions in computational tasks. In practice, solving these equations can help determine consistent solutions and avoid divergences in mathematical modelling.