Math Problem Statement
Find the eigen values of A3 – 3A2 + A where A = 4 6 6 1 3 2 -1 -4 -3
Solution
To find the eigenvalues of , where is given as:
4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -4 & -3 \end{bmatrix},$$ ### Step-by-Step Process: 1. **Eigenvalues and Powers**: Eigenvalues of a matrix power or polynomial depend on the eigenvalues of the matrix itself. Let the eigenvalues of $$A$$ be $$\lambda_1, \lambda_2, \lambda_3$$. For any matrix polynomial $$P(A)$$, the eigenvalues are $$P(\lambda_1), P(\lambda_2),$$ and $$P(\lambda_3)$$, where $$P(\lambda)$$ is the polynomial applied to the eigenvalues of $$A$$. 2. **Find Eigenvalues of $$A$$**: To determine the eigenvalues of $$A$$, solve the characteristic equation: $$\det(A - \lambda I) = 0,$$ where $$I$$ is the identity matrix. 3. **Apply Polynomial to Eigenvalues**: Once the eigenvalues $$\lambda_1, \lambda_2, \lambda_3$$ are found, compute the eigenvalues of $$A^3 - 3A^2 + A$$ using: $$\mu_i = \lambda_i^3 - 3\lambda_i^2 + \lambda_i, \quad \text{for } i = 1, 2, 3.$$ Let me calculate these values explicitly. numpy as np # Define the matrix A A = np.array([[4, 6, 6], [1, 3, 2], [-1, -4, -3]]) # Compute the eigenvalues of A eigenvalues_A = np.linalg.eigvals(A) # Compute the polynomial P(λ) = λ^3 - 3λ^2 + λ for each eigenvalue of A eigenvalues_polynomial = eigenvalues_A**3 - 3*eigenvalues_A**2 + eigenvalues_A eigenvalues_A, eigenvalues_polynomialThe eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 4, \quad \lambda_2 = 1, \quad \lambda_3 = -1.$$ The eigenvalues of $$A^3 - 3A^2 + A$$ are: $$\mu_1 = 20, \quad \mu_2 = -1, \quad \mu_3 = -5.$$ ### Final Answer: The eigenvalues of $$A^3 - 3A^2 + A$$ are $$20, -1,$$ and $$-5$$. Would you like detailed explanations for the steps? --- ### Related Questions: 1. What are eigenvectors, and how do they relate to eigenvalues in this problem? 2. How is the characteristic polynomial derived for $$A$$? 3. Can you explain the significance of matrix powers in polynomial evaluation? 4. How do matrix transformations affect eigenvalues? 5. Could $$A^3 - 3A^2 + A$$ have any geometric interpretation? ### Tip: When solving for eigenvalues of a polynomial matrix, remember that evaluating the polynomial directly on the eigenvalues of the original matrix saves computational effort!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
{"LinearAlgebra":["Eigenvalues","Eigenvectors","Matrix Polynomials"]}
Formulas
Characteristic equation: det(A - λI) = 0
Polynomial evaluation: eigenvalues (λ) = λ³ - 3λ² + 2λ
Theorems
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Suitable Grade Level
Undergraduate