Chapter 3: Problem 22

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. $$\left(\begin{array}{rrr} -3 & 2 & 2 \\ 2 & 1 & 3 \\ 2 & 3 & 1 \end{array}\right)$$

Short Answer

Expert verified

Eigenvalues: 3, -4, -5; corresponding eigenvectors can be found by solving $(A - \lambda I)x = 0$.

Step by step solution

- Write Down the Matrix

The given matrix is \[ A = \begin{pmatrix} -3 & 2 & 2 \ 2 & 1 & 3 \ 2 & 3 & 1 \end{pmatrix} \]

- Find the Characteristic Polynomial

Calculate the determinant of $A - \lambda I$. \[ A - \lambda I = \begin{pmatrix} -3 - \lambda & 2 & 2 \ 2 & 1 - \lambda & 3 \ 2 & 3 & 1 - \lambda \end{pmatrix} \]Find the determinant: \[ \text{det}(A - \lambda I) = \left| \begin{matrix} -3 - \lambda & 2 & 2 \ 2 & 1 - \lambda & 3 \ 2 & 3 & 1 - \lambda \end{matrix} \right| \]

- Calculate the Determinant

Expand the determinant along the first row: \[ \det(A - \lambda I) = (-3-\lambda)[(1-\lambda)(1-\lambda) - 9] - 2[2(1-\lambda) - 6] + 2[2\cdot3 - 2(1-\lambda)] \]Simplify each term: \[ = (-3-\lambda)[\lambda^2 - 2\lambda + 1 - 9] - 2[2 - 2\lambda - 6] + 2[6 - 2 + 2\lambda] \]

- Form Characteristic Equation

Combine and simplify: \[ = (-3-\lambda)(\lambda^2 - 2\lambda - 8) - 2(-2\lambda - 4) + 2(4 + 2\lambda) \]Further simplification: \[ = (-3-\lambda)(\lambda^2 - 2\lambda - 8) + 4\lambda + 8 + 8 + 4\lambda \]Combine like terms, characteristic polynomial: \[ P(\lambda) = \lambda^3 + 4\lambda^2 - \lambda - 60 \]

- Find Eigenvalues

Solve the characteristic polynomial $ \lambda^3 + 4\lambda^2 - \lambda - 60 = 0 $. Using the Rational Root Theorem or factoring, find initial roots possibly giving the values:$ \lambda_1 = 3 $, $ \lambda_2 = -4 $, $ \lambda_3 = -5 $

- Find Eigenvectors

For each eigenvalue, solve the system $ (A - \lambda I)x = 0 $. For $ \lambda_1 = 3 $: \[ \begin{pmatrix} -6 & 2 & 2 \ 2 & -2 & 3 \ 2 & 3 & -2 \end{pmatrix} x = 0 \] Solving this system yields: \[ x_1 = k \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} \]Do similarly for $ \lambda_2 = -4 $ and $ \lambda_3 = -5 $.

- Verify Using Computational Tools

Check the eigenvalues and eigenvectors using a computer or calculator to verify the results obtained.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Characteristic Polynomial

To find the eigenvalues of a matrix, we first need the characteristic polynomial. A matrix's characteristic polynomial is obtained by calculating the determinant of the matrix subtraction from lambda times the identity matrix, denoted as $\text{det}(A - \lambda I)$. This polynomial has the eigenvalues as roots.
For the given matrix, we form the new matrix by subtracting $\text{\lambda I}$ from the matrix A. The determinant of this new matrix is expanded and simplified to form the characteristic polynomial.
In this exercise, the characteristic polynomial was calculated to be $\lambda^3 + 4\lambda^2 - \lambda - 60$. Solving this polynomial equation will give us the eigenvalues of the matrix.

Determinant

The determinant of a matrix is a special number that can help solve systems of linear equations and find the inverse of the matrix. When finding eigenvalues, we calculate the determinant of $A - \lambda I$.
In this problem, we expanded the determinant of the resulting matrix and simplified it step by step to derive the characteristic polynomial. By expanding along the first row, each term was broken down and simplified leading us to the resulting polynomial. The steps included multiplication and combining like terms. This determinant tells us about the solutions (eigenvalues) of our original matrix.

Matrix Algebra

Matrix algebra includes operations with matrices, such as addition, multiplication, finding determinants, and working with eigenvalues and eigenvectors. Understanding these operations is key to solving problems involving eigenvalues and eigenvectors.
In this exercise, we practiced matrix subtraction, determinant calculation, and solving systems of linear equations. These are fundamental skills in matrix algebra, useful in all areas of linear transformations and vector spaces.
For example, to find eigenvectors, matrix multiplication is used to solve the equation $ (A - \lambda I)x = 0 $ where \ x \ is the eigenvector corresponding to an eigenvalue \ $ \lambda \ $.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. Eigenvalues and eigenvectors are central concepts in linear algebra, representing scale factors and directions along which linear transformations act by merely stretching or compressing.
From theoretical perspectives to practical applications in fields like engineering, computer science, and physics, understanding linear algebra's core concepts, such as characteristic polynomials and matrix operations, is crucial. In this exercise, we saw the step-by-step process of deriving eigenvalues and eigenvectors, highlighting the transition from abstract theory to practical calculation.