Math Problem Statement
Find the eigen vectors and eigen values of given matrix
A = 8 -8 -12 4 -3 -2 3 -4 1
Solution
To find the eigenvalues and eigenvectors of the matrix , we proceed as follows:
Given the matrix:
\begin{bmatrix} 8 & -8 & -12 \\ 4 & -3 & -2 \\ 3 & -4 & 1 \end{bmatrix}$$ ### Step 1: Find the Eigenvalues The eigenvalues $$\lambda$$ are found by solving the **characteristic equation**: $$\det(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix. ### Step 2: Find the Eigenvectors For each eigenvalue $$\lambda$$, solve the equation: $$(A - \lambda I)x = 0$$ to find the corresponding eigenvector(s). Let me calculate this. numpy as np # Define the matrix A A = np.array([[8, -8, -12], [4, -3, -2], [3, -4, 1]]) # Calculate eigenvalues and eigenvectors eigenvalues, eigenvectors = np.linalg.eig(A) eigenvalues, eigenvectors### Results The eigenvalues and eigenvectors of the matrix $$A$$ are: #### Eigenvalues: $$\lambda_1 = 1.828 + 5.393i, \quad \lambda_2 = 1.828 - 5.393i, \quad \lambda_3 = 2.343$$ #### Eigenvectors (corresponding to $$\lambda_1, \lambda_2, \lambda_3$$): $$\mathbf{v}_1 = \begin{bmatrix} -0.856 \\ -0.316 + 0.267i \\ -0.230 + 0.207i \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} -0.856 \\ -0.316 - 0.267i \\ -0.230 - 0.207i \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -0.795 \\ -0.606 \\ 0.029 \end{bmatrix}$$ ### Notes: - Two eigenvalues are complex, so their corresponding eigenvectors also have complex components. - One eigenvalue ($$\lambda_3$$) is real, so its eigenvector is real as well. Would you like me to provide step-by-step details on how these values were derived? --- ### Related Questions: 1. What is the physical significance of complex eigenvalues and eigenvectors? 2. How do eigenvalues relate to the determinant and trace of a matrix? 3. Can eigenvectors corresponding to complex eigenvalues be orthogonal? 4. What are some applications of eigenvalues and eigenvectors in engineering? 5. How can eigenvalues be computed manually for a $$3 \times 3$$ matrix? --- **Tip:** Eigenvalues with imaginary parts often indicate oscillatory behavior in systems like vibrations or quantum mechanics.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Formulas
det(A - λI) = 0 (Characteristic Equation)
(A - λI)x = 0 (Eigenvector Equation)
Theorems
Spectral Theorem
Cayley-Hamilton Theorem
Suitable Grade Level
Grades 11-12
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