Math Problem Statement

Solve step by step without mistake

Find the eigenvalues and eigenvectors of the matrix A = [[-1, -2, -3], [4, 5, -6], [7, -8, 9]].

Solution

Step-by-Step Solution for Eigenvalues and Eigenvectors

  1. Given Matrix AA:
-1 & -2 & -3 \\ 4 & 5 & -6 \\ 7 & -8 & 9 \end{pmatrix}$$ 2. **Eigenvalues:** The characteristic equation is derived from $$\det(A - \lambda I) = 0$$, where $$\lambda$$ represents the eigenvalues. The eigenvalues of the matrix $$A$$ are solutions to this equation. The eigenvalues for this matrix are complex, expressed as: $$\lambda_1 = 13/3 + (-1/2 - \frac{\sqrt{3} i}{2}) \cdot \alpha + \frac{133}{9 \cdot (-1/2 - \frac{\sqrt{3} i}{2}) \cdot \alpha}$$ $$\lambda_2 = 13/3 + \frac{133}{9 \cdot (-1/2 + \frac{\sqrt{3} i}{2}) \cdot \alpha} + (-1/2 + \frac{\sqrt{3} i}{2}) \cdot \alpha$$ $$\lambda_3 = \frac{133}{9 \cdot \alpha} + 13/3 + \alpha$$ Where $$\alpha = (2 \cdot \sqrt{13043} + \frac{6355}{27})^{1/3}$$. 3. **Eigenvectors:** Each eigenvalue has a corresponding eigenvector. For each $$\lambda$$, we solve the system $$(A - \lambda I) \mathbf{v} = 0$$ to find the eigenvector $$\mathbf{v}$$. The eigenvectors corresponding to the eigenvalues are expressed as matrices, each with complex entries: - For $$\lambda_1$$: $$\mathbf{v}_1 = \begin{pmatrix} \frac{2}{5} + \dots \\ \frac{14}{15} + \dots \\ 1 \end{pmatrix}$$ - For $$\lambda_2$$: $$\mathbf{v}_2 = \begin{pmatrix} \frac{2}{5} + \dots \\ \frac{14}{15} + \dots \\ 1 \end{pmatrix}$$ - For $$\lambda_3$$: $$\mathbf{v}_3 = \begin{pmatrix} \frac{2}{5} + \dots \\ \frac{14}{15} + \dots \\ 1 \end{pmatrix}$$ ### Summary: - The eigenvalues are complex numbers with real and imaginary parts. - The eigenvectors are complex as well, with expressions depending on the cube root term $$\alpha$$. Would you like further details on any of the steps? ### 5 Related Questions: 1. How do we simplify cube roots in eigenvalue calculations? 2. What is the significance of complex eigenvalues in this matrix? 3. How are eigenvectors geometrically interpreted? 4. How do eigenvalues relate to the determinant and trace of a matrix? 5. What applications can matrices with complex eigenvalues have? ### Tip: When working with complex eigenvalues and eigenvectors, it is important to carefully track real and imaginary parts to avoid errors in interpretation.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations

Formulas

Characteristic equation: det(A - λI) = 0
Eigenvalue equation: A*v = λ*v

Theorems

Eigenvalue theorem
Spectral Theorem for Matrices

Suitable Grade Level

Undergraduate

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