Question

Find the eigenvalues and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer. $$\left(\begin{array}{rrr} -1 & 1 & 3 \\ 1 & 2 & 0 \\ 3 & 0 & 2 \end{array}\right)$$

Short answer

Eigenvalues are 3, -2, and 2 with corresponding eigenvectors found by solving (A-λI)v=0.

Step-by-step solution

Write Down the Matrix

Given matrix: \[ A = \left(\begin{array}{rrr} -1 & 1 & 3 \ 1 & 2 & 0 \ 3 & 0 & 2 \end{array}\right) \]

Find the Characteristic Equation

To find the eigenvalues, solve the characteristic equation \[ \text{det}(A - \lambda I) = 0 \]. Subtract \( \lambda \) along the diagonal and find the determinant: \[ \left| \begin{array}{ccc} -1 - \lambda & 1 & 3 \ 1 & 2 - \lambda & 0 \ 3 & 0 & 2 - \lambda \end{array} \right| = 0 \].

Calculate the Determinant

Expand the determinant along the first row: \[ - ( -1 - \lambda ) \left( (2 - \lambda) (2 - \lambda) - 0 \right) + 1 \left( 1 \cdot (2 - \lambda) - 0 \right) + 3 \left( 1 \cdot 0 - (2 - \lambda) \right) = 0 \].Simplifying further:\[ -( -1 - \lambda) (4 - 4 \lambda + \lambda^2) + (2 - \lambda) - 3 (2 - \lambda) = 0 \]\[ (1 + \lambda) (\lambda^2 - 4 \lambda + 4) + (2 - \lambda) - 3 (2 - \lambda) = 0 \].

Solve the Polynomial

Simplify the polynomial:\[ \lambda^3 - 3\lambda + 4 = 0 \].Solve the cubic equation to find the eigenvalues \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\). After calculation, the eigenvalues are \[ \lambda_1 = 3, \lambda_2 = -2, \lambda_3 = 2 \].

Find Eigenvectors

For each eigenvalue, solve \( (A - \lambda I)\mathbf{v} = 0 \) to find the corresponding eigenvectors. For \( \lambda_1 = 3 \):\[ (A - 3I)\mathbf{v} = 0 \]. Solve \[ \left(\begin{array}{ccc} -4 & 1 & 3 \ 1 & -1 & 0 \ 3 & 0 & -1 \end{array}\right) \left(\begin{array}{c} v_1 \ v_2 \ v_3 \end{array}\right) = 0 \].For \( \lambda_2 = -2 \):\[ (A + 2I)\mathbf{v} = 0 \].Solve \[ \left(\begin{array}{ccc} 1 & 1 & 3 \ 1 & 4 & 0 \ 3 & 0 & 4 \end{array}\right) \left(\begin{array}{c} v_1 \ v_2 \ v_3 \end{array}\right) = 0 \].For \( \lambda_3 = 2 \):\[ (A - 2I)\mathbf{v} = 0 \]. Solve \[ \left(\begin{array}{ccc} -3 & 1 & 3 \ 1 & 0 & 0 \ 3 & 0 & 0 \end{array}\right) \left(\begin{array}{c} v_1 \ v_2 \ v_3 \end{array}\right) = 0 \].

Key concepts

Characteristic Equation

The characteristic equation is crucial in finding eigenvalues of a matrix. To determine the characteristic equation, we start with the matrix \( A \) and subtract \( \lambda\ \) (an unknown scalar) from the diagonal elements. This forms the matrix \( A - \lambda I \), where \( I \) is the identity matrix. Then, we find the determinant and set it to zero: \[ \text{det}(A - \lambda I) = 0 \]. This equation is a polynomial whose roots give us the eigenvalues.

Determinants

Determinants are scalar values that can be computed from a square matrix. They are essential in various calculations, including solving the characteristic equation. The determinant of a 3x3 matrix can be calculated using cofactor expansion. For instance, given matrix \[ A = \left(\begin{array}{ccc} -1 & 1 & 3 \ 1 & 2 & 0 \ 3 & 0 & 2 \end{array}\right) \], the determinant is found by breaking it into smaller 2x2 determinants. This process is known as expansion along a row or column. The resulting polynomial equation helps us find the eigenvalues.

Matrix Operations

Matrix operations form the basis of many linear algebra concepts, including finding eigenvalues and eigenvectors. Key operations include addition, subtraction, multiplication, and finding the inverse. In our exercise, we use the operation \( A - \lambda I \). We also employ row reduction when solving for eigenvectors. These operations manipulate the matrix into simpler forms, making it easier to apply other concepts like determinants and characteristic equations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. In this exercise, understanding eigenvalues and eigenvectors is fundamental. Eigenvalues show how much vectors are scaled during a transformation, while eigenvectors show the direction of this scaling. Together, they provide deep insights into matrix behavior, crucial for fields like physics, computer science, and engineering. Practicing problems by hand, as suggested, solidifies these concepts, allowing for better comprehension and application.