Math Problem Statement
Find eigen valus and eigen vector for a=(7 -2 0
Find the eigenvalues and eigenvectors of the matrix A = [[7, -2, 0], [-2, 6, -2], [0, -2, 5]]
Solution
The given problem is to find the eigenvalues and eigenvectors of the matrix :
7 & -2 & 0 \\ -2 & 6 & -2 \\ 0 & -2 & 5 \end{pmatrix}$$ To find the eigenvalues, we need to solve the characteristic equation, which is derived from $$\det(A - \lambda I) = 0$$, where $$I$$ is the identity matrix and $$\lambda$$ represents the eigenvalues. I'll compute the eigenvalues and eigenvectors step-by-step. Let me begin this process.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 9, \quad \lambda_2 = 6, \quad \lambda_3 = 3$$ The corresponding eigenvectors are: $$\text{For } \lambda = 9: \quad \begin{pmatrix} -\frac{2}{3} \\ \frac{2}{3} \\ -\frac{1}{3} \end{pmatrix}$$ $$\text{For } \lambda = 6: \quad \begin{pmatrix} -\frac{2}{3} \\ -\frac{1}{3} \\ \frac{2}{3} \end{pmatrix}$$ $$\text{For } \lambda = 3: \quad \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix}$$ These eigenvectors are normalized and correspond to each eigenvalue. Would you like more details or have any further questions? Here are 5 related questions: 1. How can we verify if the computed eigenvectors are correct? 2. What is the significance of eigenvalues and eigenvectors in applications? 3. Can you explain how to normalize an eigenvector? 4. How does the characteristic equation relate to eigenvalues? 5. What changes if we apply a transformation using matrix $$A$$ to these eigenvectors? **Tip:** Eigenvectors corresponding to different eigenvalues are always linearly independent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Theory
Formulas
Characteristic Equation: det(A - λI) = 0
Eigenvector Equation: (A - λI)v = 0
Theorems
Eigenvalue Theorem
Spectral Theorem for symmetric matrices (if applicable)
Suitable Grade Level
College/University level (Linear Algebra)