Problem 52 - Calculus for Biology and Medicine

Question

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

Eigenvalues are \(\lambda_1 = -1\), \(\lambda_2 = 2\); Eigenvectors are \(\textbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), \(\textbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\). Lines are \(x = t\) and \(y = t\).

Step 1: Find the Characteristic Equation

To find the eigenvalues of the matrix \(A\), calculate the characteristic equation: \( ext{det}(A - \lambda I) = 0 \). For matrix \(A = \begin{bmatrix} -1 & 0 \ 0 & 2 \end{bmatrix}\), the identity matrix \(I\) is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), so \(A - \lambda I = \begin{bmatrix} -1-\lambda & 0 \ 0 & 2-\lambda \end{bmatrix}\). The determinant is \((-1-\lambda)(2-\lambda) = 0\).

Step 2: Solve for Eigenvalues

Solving the characteristic equation \((-1-\lambda)(2-\lambda) = 0\), we find the eigenvalues: \(\lambda_1 = -1\) and \(\lambda_2 = 2\).

Step 3: Find Eigenvectors for \(\lambda_1 = -1\)

Substitute \(\lambda_1 = -1\) into \(A - \lambda I\) to find the eigenvector: \(\begin{bmatrix} 0 & 0 \ 0 & 3 \end{bmatrix} \textbf{v} = 0\). This implies \(\textbf{v} = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) is an eigenvector corresponding to \(\lambda_1 = -1\).

Step 4: Find Eigenvectors for \(\lambda_2 = 2\)

Substitute \(\lambda_2 = 2\) into \(A - \lambda I\) to find the eigenvector: \(\begin{bmatrix} -3 & 0 \ 0 & 0 \end{bmatrix} \textbf{v} = 0\). This implies \(\textbf{v} = \begin{bmatrix} 0 \ 1 \end{bmatrix}\) is an eigenvector corresponding to \(\lambda_2 = 2\).

Step 5: Determine Equations of Lines

The equations of the lines through the origin in the direction of the eigenvectors \(\textbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(\textbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\) are \(x = t\) and \(y = t\), respectively, where \(t\) is a scalar.

Step 6: Graphing Needed Components

Graph the lines through the origin in the directions of \(\textbf{v}_1\) and \(\textbf{v}_2\), along with the eigenvectors themselves: \(\textbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(\textbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\). Also, graph \(A\textbf{v}_1 = \begin{bmatrix} -1 \ 0 \end{bmatrix}\) and \(A\textbf{v}_2 = \begin{bmatrix} 0 \ 2 \end{bmatrix}\).

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that is fundamental in understanding and solving systems of linear equations. It is extensively used in a variety of fields including physics, computer science, and engineering. The core elements of linear algebra include vectors and matrices, which are essentially organized collections of numbers that represent data or coefficients in equations. By manipulating these structures, we can solve complex problems involving multiple variables. One of the pivotal concepts in linear algebra is the idea of eigenvalues and eigenvectors. These provide valuable insights into the properties of linear transformations, which can simplify complex systems by reducing them to their most fundamental components. By learning about how matrices can be decomposed in terms of their eigenvalues and eigenvectors, students can better analyze and understand linear systems.

Matrices

Matrices are rectangular arrays of numbers organized into rows and columns. They form the foundation of linear algebra and are used to represent linear transformations that can be applied to vectors. In order to work effectively with matrices, certain operations are essential, such as addition, subtraction, multiplication, and finding the determinant. The operation of finding the determinant is closely linked with determining the characteristics of a matrix, such as whether the matrix has an inverse.

Matrices can also be categorized based on their properties, such as being square (having the same number of rows and columns) or diagonal (non-diagonal elements are zero). These specific types can substantially simplify matrix computations. When dealing with problems like finding eigenvalues and eigenvectors, understanding how to manipulate matrices and how to calculate their determinants is crucial. Being adept at these operations helps explain complex systems and make predictions about their behavior through transformations.

Characteristic Equation

The characteristic equation is a fundamental tool used for finding eigenvalues of a matrix. It is derived from a square matrix, usually denoted by subtracting \(\lambda I\) from the matrix and then setting the determinant of the result to zero. This equation helps identify the eigenvalues, which are key to understanding the matrix's effect on vectors. The equation appears as a polynomial where the roots correspond to the eigenvalues.

Solving the characteristic equation can sometimes be challenging, particularly for larger matrices. However, it results in a polynomial equation, typically quadratic for 2x2 matrices, and each solution corresponds to a potential eigenvalue. Recognizing these solutions aids in subsequent steps of linear algebra problems, such as determining eigenvectors. Understanding how to construct and solve a characteristic equation is vital as it plays a critical role in simplifying matrices and understanding the transformations they represent.