Math Problem Statement
Find the eigen values and eigen vector of the following matrix A = 3 2 4 2 0 2 4 2 3
Solution
To find the eigenvalues and eigenvectors of matrix , follow these steps:
Given matrix :
3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$$ ### Step 1: Find the eigenvalues Eigenvalues are obtained by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Where: - $$A$$ is the matrix, - $$\lambda$$ is the eigenvalue, - $$I$$ is the identity matrix of the same size as $$A$$. The matrix $$A - \lambda I$$ is: $$A - \lambda I = \begin{pmatrix} 3 - \lambda & 2 & 4 \\ 2 & 0 - \lambda & 2 \\ 4 & 2 & 3 - \lambda \end{pmatrix}$$ Now, compute the determinant of this matrix: $$\text{det}(A - \lambda I) = \begin{vmatrix} 3 - \lambda & 2 & 4 \\ 2 & -\lambda & 2 \\ 4 & 2 & 3 - \lambda \end{vmatrix}$$ Expanding this determinant will give the characteristic polynomial, and solving this for $$\lambda$$ will yield the eigenvalues. ### Step 2: Find the eigenvectors Once the eigenvalues are determined, the eigenvectors can be found by solving the system: $$(A - \lambda I) \vec{v} = 0$$ For each eigenvalue $$\lambda$$, we substitute it back into $$A - \lambda I$$ and solve for the vector $$\vec{v}$$ (the eigenvector) that satisfies this equation. I'll now compute the determinant to find the eigenvalues and then determine the eigenvectors.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = -1, \quad \lambda_2 = 8, \quad \lambda_3 = -1$$ The corresponding eigenvectors are: $$\vec{v_1} = \begin{pmatrix} -0.7454 \\ 0.2981 \\ 0.5963 \end{pmatrix}, \quad \vec{v_2} = \begin{pmatrix} 0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad \vec{v_3} = \begin{pmatrix} -0.2076 \\ -0.7760 \\ 0.5956 \end{pmatrix}$$ ### Summary: - Eigenvalues: $$-1, 8, -1$$ - Eigenvectors: Listed above. Would you like a detailed step-by-step breakdown of the calculations? Here are 5 follow-up questions related to eigenvalues and eigenvectors: 1. How do the eigenvectors relate to the original matrix transformation? 2. What is the geometric significance of the eigenvalues? 3. How do you normalize an eigenvector? 4. What is the characteristic equation for this matrix? 5. Can a matrix have complex eigenvalues? **Tip**: Eigenvalues tell you the factor by which the transformation scales a vector, while eigenvectors show the directions along which this scaling occurs.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Determinants
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Theorems
Eigenvalue theorem
Determinant property in matrices
Suitable Grade Level
Grades 11-12, Undergraduate
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