Math Problem Statement
Find the eigen values and eigen vectors of following matrix
4 2 -2 -5 3 2 -2 4 1
Solution
To find the eigenvalues and eigenvectors of the given matrix:
4 & 2 & -2 \\ -5 & 3 & 2 \\ -2 & 4 & 1 \end{bmatrix}$$ ### Steps: 1. **Eigenvalues**: The eigenvalues $$\lambda$$ satisfy the characteristic equation: $$\det(A - \lambda I) = 0$$ Here, $$I$$ is the identity matrix of the same size as $$A$$. $$A - \lambda I = \begin{bmatrix} 4 - \lambda & 2 & -2 \\ -5 & 3 - \lambda & 2 \\ -2 & 4 & 1 - \lambda \end{bmatrix}$$ The determinant of $$A - \lambda I$$ is calculated as: $$\det(A - \lambda I) = \begin{vmatrix} 4 - \lambda & 2 & -2 \\ -5 & 3 - \lambda & 2 \\ -2 & 4 & 1 - \lambda \end{vmatrix}$$ Expanding this determinant results in a cubic equation in $$\lambda$$. 2. **Eigenvectors**: For each eigenvalue $$\lambda$$, solve the system $$(A - \lambda I)\mathbf{x} = 0$$ to find the eigenvector(s) $$\mathbf{x}$$. I will calculate the eigenvalues and eigenvectors explicitly. sympy as sp # Define the matrix A A = sp.Matrix([ [4, 2, -2], [-5, 3, 2], [-2, 4, 1] ]) # Compute the eigenvalues and eigenvectors eigen_data = A.eigenvects() eigen_dataThe eigenvalues and corresponding eigenvectors of the matrix are as follows: 1. **Eigenvalue** $$\lambda = 1$$: - Eigenvector: $$\mathbf{x}_1 = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{4} \\ 1 \end{bmatrix}$$ 2. **Eigenvalue** $$\lambda = 2$$: - Eigenvector: $$\mathbf{x}_2 = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2} \\ 1 \end{bmatrix}$$ 3. **Eigenvalue** $$\lambda = 5$$: - Eigenvector: $$\mathbf{x}_3 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$$ These results show that the matrix has three distinct eigenvalues, each with a corresponding eigenvector. --- Would you like detailed steps for how the eigenvalues and eigenvectors were calculated? Here are five questions you might want to explore next: 1. How is the characteristic polynomial derived in detail? 2. What does the geometric interpretation of eigenvectors and eigenvalues look like? 3. Can eigenvalues be used to diagonalize the matrix $$A$$? 4. What does the determinant of the matrix tell us about its eigenvalues? 5. How do eigenvalues relate to the stability of a system in applied problems? **Tip:** Eigenvectors can be scaled by any non-zero scalar. They are typically normalized to make computations easier.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Determinants
Formulas
Characteristic equation: det(A - λI) = 0
Eigenvalue equation: A * x = λ * x
Theorems
Eigenvalue theorem
Determinant property for eigenvalues
Suitable Grade Level
Undergraduate (Linear Algebra Course)
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