Math Problem Statement
Find the eigenvectors for matrix A = [[1, 1, 1], [-1, -3, -3], [2, 4, 4]]
Solution
We are tasked with finding the eigenvectors of matrix , where:
1 & 1 & 1 \\ -1 & -3 & -3 \\ 2 & 4 & 4 \end{pmatrix}$$ ### Steps to Find the Eigenvectors: 1. **Find the eigenvalues**: We start by solving the characteristic equation: $$\det(A - \lambda I) = 0$$ where $$I$$ is the identity matrix and $$\lambda$$ represents the eigenvalues. 2. **Find the eigenvectors**: For each eigenvalue $$\lambda$$, solve the system: $$(A - \lambda I) \mathbf{v} = 0$$ to find the eigenvector $$\mathbf{v}$$. Let me calculate this step by step.The eigenvalues of matrix $$A$$ are approximately: $$\lambda_1 = 2, \quad \lambda_2 \approx 0, \quad \lambda_3 \approx 0$$ The corresponding eigenvectors (each column corresponds to an eigenvector) are: $$\mathbf{v}_1 = \begin{pmatrix} 0.2673 \\ -0.5345 \\ 0.8018 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -5.86 \times 10^{-9} \\ 0.7071 \\ -0.7071 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} -5.86 \times 10^{-9} \\ -0.7071 \\ 0.7071 \end{pmatrix}$$ These are the eigenvectors corresponding to the eigenvalues $$2$$, $$0$$, and $$0$$, respectively. Would you like more details or further clarifications? ### Related Questions: 1. How do we verify if a vector is an eigenvector of a matrix? 2. What is the physical significance of eigenvalues and eigenvectors? 3. How do eigenvalues help in diagonalizing a matrix? 4. What is the characteristic polynomial of a matrix, and how is it related to eigenvalues? 5. Can a matrix have complex eigenvalues and eigenvectors? **Tip:** Eigenvectors corresponding to distinct eigenvalues are linearly independent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Theory
Formulas
det(A - λI) = 0
(A - λI)v = 0
Theorems
Eigenvalue Theorem
Determinant Properties
Suitable Grade Level
Undergraduate (Linear Algebra Course)