Math Problem Statement
Eigen vector
Eigen vector
Solution
The uploaded image contains a 3x3 matrix labeled as , given as:
6 & -2 & 2 \\ 2 & -3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$$ You are asking about the **eigenvectors** of this matrix. To compute them, we first need to calculate the eigenvalues, which are determined by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Once the eigenvalues are found ($$\lambda$$), the eigenvectors can be computed by solving $$(A - \lambda I)\mathbf{v} = 0$$ for each eigenvalue $$\lambda$$, where $$\mathbf{v}$$ is the eigenvector. Let me calculate the eigenvalues and eigenvectors for this matrix. One moment! numpy as np # Define the matrix A A = np.array([[6, -2, 2], [2, -3, -1], [2, -1, 3]]) # Compute eigenvalues and eigenvectors eigenvalues, eigenvectors = np.linalg.eig(A) eigenvalues, eigenvectorsThe eigenvalues and eigenvectors of the matrix $$A$$ are as follows: ### Eigenvalues: 1. $$\lambda_1 = 6.6904$$ 2. $$\lambda_2 = -2.6904$$ 3. $$\lambda_3 = 2.0000$$ ### Corresponding Eigenvectors: Each column of the eigenvector matrix corresponds to an eigenvalue: $$\text{Eigenvector for } \lambda_1 = 6.6904: \begin{bmatrix} 0.8860 \\ 0.1371 \\ 0.4430 \end{bmatrix}$$ $$\text{Eigenvector for } \lambda_2 = -2.6904: \begin{bmatrix} -0.2011 \\ -0.9744 \\ -0.1006 \end{bmatrix}$$ $$\text{Eigenvector for } \lambda_3 = 2.0000: \begin{bmatrix} -0.5571 \\ -0.3714 \\ 0.7428 \end{bmatrix}$$ Let me know if you'd like further explanation on any step, or help with normalization or interpretation! --- ### Follow-up Questions: 1. What does it mean for a matrix to have eigenvalues and eigenvectors? 2. How are eigenvectors used in practical applications (e.g., physics or machine learning)? 3. How do eigenvalues determine if a matrix is invertible or not? 4. Would you like the eigenvectors written in terms of a basis for the solution space? 5. Shall I provide steps for solving the characteristic polynomial? ### Tip: Eigenvectors corresponding to distinct eigenvalues are linearly independent and form a basis for the vector space if the matrix is diagonalizable.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector computation: (A - λI)v = 0
Theorems
Eigenvalue and Eigenvector Theorem
Properties of Determinants
Linear Independence of Eigenvectors
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)
Related Recommendation
Finding Eigenvalues and Eigenvectors of a 3x3 Matrix
Eigenvalues and Eigenvectors of 3x3 Matrix A=[[3,2,2],[2,3,2],[3,-3,-3]]
Find Eigenvalues and Eigenvectors for Matrix A = [[8, -6, 2], [-6, 7, -4], [2, -4, 3]]
Find Eigenvalues and Eigenvectors of a 3x3 Matrix
Finding Eigenvalues and Eigenvectors of a 3x3 Matrix