Question

In each exercise, \(\lambda\) is an eigenvalue of the given matrix \(A\). Determine an eigenvector corresponding to \(\lambda\). $$ =\left[\begin{array}{rr} 5 & 3 \\ -4 & -3 \end{array}\right], \quad \lambda=-1 $$

Short answer

Based on the given step by step solution, the eigenvector corresponding to the eigenvalue \(\lambda=-1\) is represented by the general solution: $$ x=t\left[\begin{array}{c} -1 \\ 2 \end{array}\right], \quad t\in \mathbb{R} $$ where \(t\) is a scalar multiplier from the set of real numbers.

Step-by-step solution

Rewrite the matrix (A-\lambda I)

Since \(\lambda=-1\), we need to find the matrix \(A-\lambda I$$=A-(-1)I=A+I\) which can be calculated as follows: $$ A+I=\left[\begin{array}{rr} 5 &3 \\\ -4 & -3 \end{array}\right]+ \left[\begin{array}{rr} 1& 0 \\ 0 & 1 \end{array}\right]= \left[\begin{array}{rr} 6 & 3 \\ -4 & -2 \end{array}\right] $$

Solve the linear system \((A-\lambda I)x=0\) using Gaussian elimination

Now we need to solve the linear system \((A-\lambda I)x=0\) or in our case \((A+I)x=0\). Start by setting an augmented matrix and apply Gaussian elimination: $$ \left[\begin{array}{cc|c} 6 & 3 & 0 \\ -4 & -2 & 0 \end{array}\right] $$ Divide the first row by 2: $$ \left[\begin{array}{cc|c} 3 & 3/2 & 0 \\ -4 & -2 & 0 \end{array}\right] $$ Add \(4/3\) times the first row to the second row: $$ \left[\begin{array}{cc|c} 3 & 3/2 & 0 \\ 0 & 0 & 0 \end{array}\right] $$ Now we have only one equation: \(3x_1+\frac{3}{2}x_2=0\). Solving for \(x_1\), we get \(x_1=-\frac{1}{2}x_2\). Let \(x_2=2t\), where \(t\) is a free parameter. Then, \(x_1=-t\). Therefore, the general solution representing the eigenvectors of \(\lambda=-1\) is: $$ x=t\left[\begin{array}{c} -1 \\ 2 \end{array}\right], \quad t\in \mathbb{R} $$

Key concepts

Eigenvalue

Eigenvalues are a fundamental concept in linear algebra, particularly when it comes to understanding the behavior of linear transformations represented by matrices. An eigenvalue, typically denoted by \( \lambda \), is a special scalar associated with a linear transformation. When a non-zero vector \( v \) is transformed by a matrix \( A \), and the result is a scalar multiple of \( v \), that scalar is the eigenvalue, and \( v \) is called an eigenvector.

Mathematically, this relationship is described by the equation \( Av = \lambda v \). In the context of the given exercise, \( \lambda = -1 \) is known, and it's our goal to find a corresponding eigenvector. An eigenvector associated with \( \lambda = -1 \) will not change direction when the transformation by matrix \( A \) is applied, but it may be stretched or compressed if \( \lambda \) is not equal to 1.

Gaussian Elimination

Gaussian elimination is a systematic method used in linear algebra to solve systems of linear equations. It involves performing row operations on the augmented matrix of a system to bring it into its row-echelon form and then solving for the variables. The three types of row operations are: swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.

Gaussian elimination can reveal if a system has a unique solution, infinitely many solutions, or no solution. In the context of finding an eigenvector, Gaussian elimination helps reduce the system \( (A-\lambda I)x=0 \) to a simpler form from which eigenvectors can be extracted. In our exercise, Gaussian elimination reduces the matrix to a form that shows us a relationship between the variables, namely that \( x_1 \) is proportional to \( x_2 \), which leads directly to finding the eigenvectors.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. Beyond solving systems of linear equations, linear algebra encompasses the study of vectors, vector spaces, linear transformations, and systems of linear inequalities.

Applications of linear algebra are vast and include areas such as computer graphics, engineering, physics, and more abstract fields like quantum mechanics. In our exercise, we employ linear algebra to identify eigenvectors and eigenvalues, showcasing its importance in both theoretical and practical applications.

Differential Equations

Differential equations are equations that relate functions to their derivatives. In the context of linear algebra, linear differential equations contain terms that are either constant or linear functions of the variable we are differentiating with respect to. These equations are essential in modeling a wide variety of real-world phenomena, including physics, engineering, economics, and biology.

Eigenvalues and eigenvectors have a significant role in solving systems of differential equations, particularly when dealing with linear systems. They help determine the behavior of the system's solutions over time. For example, negative eigenvalues can signal that the system will eventually stabilize at an equilibrium point. While differential equations were not directly part of this exercise, understanding their relationship with eigenvalues and eigenvectors provides an appreciation for the depth and utility of these concepts in various scientific and mathematical applications.