Math Problem Statement
1 3 3 1 4 3 -1 3 4 Find eigen values and vectors
Solution
To find the eigenvalues and eigenvectors of the matrix , follow these steps:
Given matrix:
1 & 3 & 3 \\ 1 & 4 & 3 \\ -1 & 3 & 4 \end{pmatrix}$$ ### Step 1: Find the Eigenvalues The eigenvalues $$\lambda$$ are found by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix and $$\lambda$$ is the eigenvalue. 1. Subtract $$\lambda$$ from the diagonal elements of $$A$$: $$A - \lambda I = \begin{pmatrix} 1-\lambda & 3 & 3 \\ 1 & 4-\lambda & 3 \\ -1 & 3 & 4-\lambda \end{pmatrix}$$ 2. Calculate the determinant: $$\text{det}(A - \lambda I) = \text{det} \begin{pmatrix} 1-\lambda & 3 & 3 \\ 1 & 4-\lambda & 3 \\ -1 & 3 & 4-\lambda \end{pmatrix}$$ Expanding this determinant: $$(1-\lambda) \left( (4-\lambda)(4-\lambda) - 3 \times 3 \right) - 3 \left( 1(4-\lambda) - 3(-1) \right) + 3 \left( 1 \times 3 - 1 \times (-1) \right)$$ Simplify and solve for $$\lambda$$. ### Step 2: Find the Eigenvectors For each eigenvalue $$\lambda_i$$, solve: $$(A - \lambda_i I) \mathbf{v} = 0$$ Where $$\mathbf{v}$$ is the eigenvector associated with $$\lambda_i$$. ### Calculation: Let me calculate the eigenvalues and eigenvectors.### Eigenvalues: The eigenvalues of the matrix $$A$$ are approximately: $$\lambda_1 = 1 + 2.95 \times 10^{-8}i, \quad \lambda_2 = 1 - 2.95 \times 10^{-8}i, \quad \lambda_3 = 7$$ Here, $$\lambda_1$$ and $$\lambda_2$$ are complex conjugates with very small imaginary parts, so they are almost real. ### Eigenvectors: The corresponding eigenvectors are: For $$\lambda_1 = 1 + 2.95 \times 10^{-8}i$$: $$\mathbf{v}_1 \approx \begin{pmatrix} -1.57 \times 10^{-16} - 2.09 \times 10^{-8}i \\ -0.7071 \\ 0.7071 \end{pmatrix}$$ For $$\lambda_2 = 1 - 2.95 \times 10^{-8}i$$: $$\mathbf{v}_2 \approx \begin{pmatrix} -1.57 \times 10^{-16} + 2.09 \times 10^{-8}i \\ -0.7071 \\ 0.7071 \end{pmatrix}$$ For $$\lambda_3 = 7$$: $$\mathbf{v}_3 \approx \begin{pmatrix} 0.5721 \\ 0.6674 \\ 0.4767 \end{pmatrix}$$ These eigenvectors are normalized (each has a magnitude of 1). Would you like further details on the steps, or do you have any questions? ### Related Questions: 1. What is the significance of eigenvalues in a matrix? 2. How are eigenvectors used in applications such as Principal Component Analysis (PCA)? 3. Can you explain how the characteristic polynomial is derived from a matrix? 4. What are some real-world applications of eigenvalues and eigenvectors? 5. How do complex eigenvalues affect the stability of a system? 6. What is the geometric interpretation of eigenvectors and eigenvalues? 7. How can you verify if a vector is an eigenvector of a matrix? 8. What happens if a matrix has repeated eigenvalues? **Tip:** Always ensure eigenvectors are normalized when using them in applications, as their direction is often more important than their magnitude.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Characteristic Equation
Determinant
Formulas
Characteristic equation
Eigenvalue calculation
Eigenvector calculation
Theorems
Spectral theorem
Suitable Grade Level
College Level
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