Math Problem Statement
find eigenvalues and eigenvectors of [7 0 "-3" "-9" "-2" 3 18 0 "-8]"
Solution
Let's find the eigenvalues and eigenvectors of the matrix:
7 & 0 & -3 \\ -9 & -2 & 3 \\ 18 & 0 & -8 \end{pmatrix}$$ ### Step 1: Find the Eigenvalues The eigenvalues of a matrix are found by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Where $$\lambda$$ is the eigenvalue and $$I$$ is the identity matrix. For matrix $$A$$, we have: $$A - \lambda I = \begin{pmatrix} 7 - \lambda & 0 & -3 \\ -9 & -2 - \lambda & 3 \\ 18 & 0 & -8 - \lambda \end{pmatrix}$$ The determinant is: $$\text{det}(A - \lambda I) = \text{det}\left(\begin{pmatrix} 7 - \lambda & 0 & -3 \\ -9 & -2 - \lambda & 3 \\ 18 & 0 & -8 - \lambda \end{pmatrix}\right)$$ We will calculate this determinant to find the characteristic polynomial and then solve for $$\lambda$$ (the eigenvalues). ### Step 2: Solve for Eigenvectors Once the eigenvalues are found, we solve $$(A - \lambda I) \mathbf{v} = 0$$ to find the corresponding eigenvectors. Let's calculate these.### Results: 1. **Eigenvalues**: The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = -2, \quad \lambda_2 = 1, \quad \lambda_3 = -2$$ 2. **Eigenvectors**: The corresponding eigenvectors are: For $$\lambda_1 = -2$$: $$\mathbf{v}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ For $$\lambda_2 = 1$$: $$\mathbf{v}_2 = \begin{pmatrix} 0.408 \\ -0.408 \\ 0.816 \end{pmatrix}$$ For $$\lambda_3 = -2$$: $$\mathbf{v}_3 = \begin{pmatrix} 0.300 \\ -0.318 \\ 0.899 \end{pmatrix}$$ These results give you the eigenvalues and the corresponding eigenvectors for the matrix $$A$$. Would you like more details on any part of the solution, or do you have any questions? ### Related Questions: 1. How are eigenvalues and eigenvectors used in solving systems of differential equations? 2. What is the geometric interpretation of eigenvectors and eigenvalues? 3. How can the diagonalization of a matrix be related to its eigenvalues and eigenvectors? 4. What happens if a matrix has complex eigenvalues? 5. How can eigenvectors be normalized? ### Tip: When computing eigenvalues and eigenvectors, always check if the matrix is diagonalizable to simplify your calculations.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Eigenvalues
Eigenvectors
Matrix Determinant
Characteristic Equation
Formulas
Characteristic equation det(A - λI) = 0
Theorems
-
Suitable Grade Level
Advanced High School